Proof that lifts of geodesics are quasi-geodesics (relatively hyperbolic groups) $\DeclareMathOperator\Cay{Cay}$Suppose $G$ is a relatively hyperbolic group with peripheral subgroups $P_1,P_2,\dots, P_n$, and suppose $\mathcal{S}$ is a finite generating set for $G$. Let $X=\Cay(G,\mathcal{S})$ denote the Cayley graph of $G$ with respect to $\mathcal{S}$, and let $\hat{X} = \Cay(G,\mathcal{S}\cup \mathcal{P})$ where $\mathcal{P} = \bigcup_{i=1}^{n} P_i$. I have seen a number of places (e.g. [3, Prop 1.14]) that there are constants $\lambda, \epsilon$ such that given a geodesic $\gamma$ in $\hat{X}$, any lift $\tilde{\gamma}$ of $\gamma$ is a $(\lambda,\epsilon)$-quasi-geodesic in $X$. Here a lift refers to a path in $X$ where one substitutes any edges of $\gamma$ that were labeled with some element of some $P_i$ with a geodesic in the corresponding left coset. This is the definition used in [3], but sometimes (such as in [1] and [2]) it looks like we just replace $\mathcal{P}$-edges of $\gamma$ with geodesics in $X$ rather than using geodesics in left cosets. (Note: I think lifts are sometimes also referred to as de-electrifications).
I have not been able to find a proof of this statement anywhere. All I have been able to find is brief outlines of how one might prove the statement (e.g. [3] mentions using Lemma 8.8 from [2] but doesn't say much more than that). Others have suggested that one might be able to deduce the proof from Theorem 1.12 parts (3) and (4) from [1], or their proofs. It is not obvious to me using any of these methods how one deduces the proof. It's also not obvious to me which definition of a lift is the one you want to use, or if it matters.
If anyone has a reference that does prove that lifts of geodesics are quasi-geodesics, or knows how one would prove this, I would greatly appreciate the help!
[1] Druţu, Cornelia; Sapir, Mark, Tree-graded spaces and asymptotic cones of groups. (With an appendix by Denis Osin and Mark Sapir)., Topology 44, No. 5, 959-1058 (2005). ZBL1101.20025.
[2] Hruska, G. Christopher, Relative hyperbolicity and relative quasiconvexity for countable groups., Algebr. Geom. Topol. 10, No. 3, 1807-1856 (2010). ZBL1202.20046.
[3] Sisto, Alessandro, Projections and relative hyperbolicity., Enseign. Math. (2) 59, No. 1-2, 165-181 (2013). ZBL1309.20036.
 A: This was not originally an answer to the question you asked but to the converse question about geodesic paths in $X$ mapping to quasigeodesic paths in $\widehat{X}$. But you asked for a reference for that in a comment, so here it is. But note that this result involves extending the given generating set of $G$ to a larger finite generating set by adjoining some elements from the parabolic subgroups.
$\mathbf{Edit}$: It think that it is not difficult to deduce the result that you want from this one, but again with the extended generating set of $G$. See the end of this post for a sketch proof.
The reference is:
Y. Antolin and L. Ciobanu,
Finite generating sets of relatively hyperbolic groups and
applications to geodesic languages.
${\mathit Trans. Amer. Math. Soc.}$, 368 (11):7965-8010, 2016.
Lemma 5.3 of that paper states:
Let $G$ be a finitely generated group hyperbolic with respect to a family of subgroups $\{H_\omega\}_{\omega \in \Omega}$, and let $Y$ be a finite generating set.
Then there exist $\lambda \ge 1$, $c \ge 0$ and a finite subset $\mathcal{H}'$ of $\mathcal{H} := \cup_{\omega \in \Omega}(H_\omega - \{1\})$ such that, for every finite generating set $X$ of $G$ with
$$Y \cup \mathcal{H}' \subseteq X \subseteq Y \cup \mathcal{H},$$
there is a finite subset $\Phi$ of non-geodesic words over $X$ satisfying: if a word $W \in X^*$ has no parabolic shortenings and does not contain subwords in $\Phi$, then the word $\widehat{W} \in (X \cup \mathcal{H})^*$ is a 2-local geodesic $(\lambda,c)$-quasigeodesic without vertex backtracking.
In particular, for every $\omega \in \Omega$ and $h \in H_\omega$, $|h|_X= |h|_{X \cap H_\omega}$.
"contains no parabolic shortenings" just means that subwords with all generators in the same parabolic subgroup $H_\omega$ are geodesics as words over $X \cap H_\omega$.
You would have to look at the paper for a precise definition of $\widehat{W}$ from $W$, but it basically means replace subwords in which all generators lie in the same $H_\omega$ by the corresponding element of $H_\omega$.
$\mathbf{Added\ later}$: Here is a rough idea of how to deduce the result you are looking for from this result.
I will denote by $\widehat{V}$ your original word that labels a geodesic path in $\widehat{X}$, and by $V$ its lift to a word over $X$ defined by substituting geodesic words over $X \cap H_\omega$ for the letters of $\widehat{V}$ that lie in $H_\omega$.
Let $W$ be a geodesic word over $X$ representing the same group element as $V$. Then, by the result above, the derived word $\widehat{W}$ labels a $(\lambda,c)$-quasigeodesic path in $\widehat{X}$ for some constants $\lambda$ and $c$.
Now by the Bounded Coset Penetration Property applied to $\widehat{V}$ and $\widehat{W}$, there is a constant $K$ such that the parabolic components $\alpha$ (i.e. maximal subwords over $X \cap H_\omega$ for some $\omega$) of $V$ have corresponding parabolic components $\alpha'$ in $W$, for which $|\ell(\alpha) - \ell(\alpha')|$ is bounded by a constant.
Since the total length of the remainder of the word $W$ is bounded by $K$ times the length of $\widehat{V}$, it follows easily that the total length $\ell(V)$ of $V$ is bounded by a constant times $\ell(W)$.
Of course to show that $V$ is quasigeodesic, you have to prove the same for each of its subwords, and you have the added technical complication that subwords might split parabolic components, but I don't think dealing with that would be problematic.
A: Here's a sketch of how you can prove this. It uses my personal favourite construction for relatively hyperbolic groups: combinatorial horoballs.
Roughly the construction to take a parabolic group $P$, here viewed as a discrete set of points. Take the cartesian product $P \times [0,\infty)$, attach $P \times [0,\infty)$ to the Cayley graph by identifying $p \sim (p,0), \forall p \in P$. Then at each level $P\times\{n\}, n \in \mathbb Z_{\geq 0}$ add length 1 edges do contract distances (see the paper for details.) Call this graph a combinatorial horoball, the complete subgraph containing $P\times\{n\}$ us called the horosphere at depth $n$ and the complete subgraph containing $P \times \{n,n+1,n+2,\ldots\}$ is called the depth $n$ horoball.
Equivariantly attach horoballs to every coset of $P$ and repeat the process for each other conjugacy class of maximal parabolic groups. Call this the cusped Cayley graph. See Dehn Fillingin elatively hyperbolic groups by Groves and Manning (also available on arXiv) for details as well as a proof that hyperbolicity of a cusped Cayley graph is equivalent to relative hyperbolicity.
Cusped Cayley graphs let you interpolate between the standard space $X$ and the electrified space $\hat X$ as follows: if you chop off the horoballs at depth $n$ then you get something q.i. to $X$ (use Svarc-Milnor). If you squeeze horoballs at depth $n$ (i.e. identify them to points) you get something q.i. to the electrified space. In fact, $\hat X$ is obtained by squeezing all horoballs of depth 1, but you can squeeze deeper balls to get spaces q.i. to $\hat X$ that are "closer" to the cusped space.
The argument goes like this. Pick $n$ deep enough so that no geodesic $\gamma$ in the cusped space will ever enter a horoball of depth $n$ then exit the horoball, then re-enter the horoball in the cusped space. To prove the existence of such $n$ requires an examination of the combinatorial horoball construction. Squeeze all the horoballs balls of depth $n+1$ to points to get $\hat X_n$ which is q.i. to $\hat X$ (this you can show directly.) There is also an $N$ and a hyperbolicity constant $\delta$ such that for all $n>N$, $\delta$ is a hyperbolicity constant for the cusped space and all squeezed spaces $\hat X_n$.
Pick $n$ much larger than $\delta$.
Next, you chop off the (image of) the depth $n+1$ horosphere in $\hat X_{n+1}$ to get $X_n$ which is q.i. to $X$.  Now when passing to to $X_{n}$ (which involves chopping off points) a geodesic $\gamma \in \hat X_{n+1}$ will get cut into pieces. You can "reconstruct" $\gamma$ by joining the severed bits with geodesics and these connecting geodesics will have to remain in the (remnants of the) level $n$ horoballs (otherwise you would have contradicted the choice of $n$) and it's easy to see that this reconstructed $\gamma$ is a long local geodesic. This is because away from the depth $n$ horoballs, the geometries of $\hat X_n$ and of $X_n$ are similar (remember that lengths of paths is literally counting edges) and otherwise, deep in a horoball, the geometry is also controlled.
What has been sketched is that for some $n$ any geodesic $\gamma$ in $\hat X_n$ is projected or can be lifted to a geodesic via the maps $$
\hat X \leftarrow \hat X_n \rightarrow X_n \stackrel{q.i.}{\rightarrow} X
$$
The result can then be obtained using stability results for quasi geodesics in hyperbolic spaces.
