This is a cross-post from a MSE question which received no answers. Beware that the notation here is a little different.

Consider the following lifting problem(s):

$\require{AMScd}$
\begin{CD}
& & & & E\\
& & & @VV{p}V\\
Y @>{g}>> X @>{f}>> B
\end{CD}
Let's work with the following assumptions and notations (**edited the indexing error pointed out by Aleksandar Milivojevic**):

- The map $p: E \rightarrow B$ is a Hurewicz fibration with path-connected base $B$ and $(d-2)$-connected fiber $F$ for some $d \geq 2$. In case $d=2$ we require $\pi_1(F)$ to be abelian.
- For every $k \geq 0$, the action of the fundamental group $\pi_1(F)$ on the homotopy group $\pi_k(F)$ is trivial. As a consequence, there is a well-defined action [Davis-Kirk, Proposition 6.62] of $\pi_1(B)$ on $\pi_k(F)$.
- $X,Y$ are finite CW complexes, $g$ is a cellular map, and $\dim(X) = d < \dim(Y)$ with the same $d$ in (1).
- For each $k \geq 0$, write $\rho(f,k): \pi_1(X) \rightarrow Aut(\pi_k(F))$ for the induced action of $\pi_1(X)$ from (2). Consider this as a local coefficient system over $X$, so that there are well-defined cohomology groups $H^*(X;\pi_k(F)_{\rho(f,k)})$ with local coefficients. There is a similar situation with $\rho(fg,k) : \pi_1(Y) \rightarrow Aut(\pi_k(F))$.

Under these conditions, there is a well-understood obstruction theory. Using [Davis-Kirk, Theorem 7.37] and the remarks later: First of all, $f$ can be lifted over the $(d-1)$-skeleton $X_{d-1}$. Second, no matter which lift over the $(d-1)$-skeleton we choose, the obstruction class for lifting it further over the $d$-skeleton is **unique**. This **primary obstruction** $z_f$ is an element of $H^d(X;\pi_{d-1}(F)_{\rho(f,d-1)})$. For dimension reasons $z_f$ is the **only** obstruction for lifting $f$ along $p$.

The problem of lifting $fg$ along $p$ has a similar obstruction theory to above. There is again a primary obstruction $z_{fg} \in H^d(Y;\pi_{d-1}(F)_{\rho(fg,d-1)})$, but now there might be higher obstructions lying in $H^{k+1}(Y;\pi_k(F)_{\rho(fg,k)})$ for $k \geq d$.

Now suppose $z_f \neq 0$ but $z_{fg} = 0$. In other words (by the naturality of the obstruction classes), $z_f$ lies in the kernel of the induced map $$g^*: H^d(X;\pi_{d-1}(F)_{\rho(f,d-1)}) \rightarrow H^d(Y;\pi_{d-1}(F)_{\rho(fg,d-1)}) \, .$$

Does this vanishing imply that $fg$ can be lifted over the whole $Y$ (not just its $d$-skeleton)? If not, can we at least say that there is a unique nonzero higher obstruction class for lifting $fg$?

My intuition is that the lifting problem for $fg$ ** should not be harder ** than the lifting problem for $f$.

*Davis, James F.; Kirk, Paul*, Lecture notes in algebraic topology, Graduate Studies in Mathematics. 35. Providence, RI: AMS, American Mathematical Society. xvi, 367 p. (2001). ZBL1018.55001.