Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that $V$ is crystalline with distinct Hodge-Tate weights $k_1 < k_2 < \dotsb < k_n$. We *do not* assume any condition on the irreducibility/semi-simplicity of $V$. In fact, we assume that we have a $G$-stable filtration $V^- \subset V$ such that $V^-$ contains all of the non-negative Hodge-Tate weights (the cyclotomic character has weight $-1$, by the way). Note: everything happening is in characteristic zero. Such a $V$ might be said to satisfy the Panchiskin Condition.

Let $A$ be a local Artin $L$-algebra with residue field $L$ (a morphism between two such rings should induce the identity on $L$). We consider the of formal deformation functor $X(A)$ of $V$ to $A$. Inside, we define a subfunctor $X^- \subset X$ with the property that a deformation $V_A$ of $V$ to $A$ is in $X^-(A)$ if and only if it has a Galois stable filtration $V_A^{-} \subset V_A$ (a free $G$-stable $A$-submodule with free quotient $V_A/V_A^{-}$) and the isomorphism $V_A\otimes_A L \simeq V$ induces an isomorphism $V_A^{-}\otimes_L A \simeq V^-$.

I believe under some reasonable hypotheses the functor $X^-$ is (relatively) representable (over $X$). Here representable means "pro-representable". For example, if $V^-$ and $V/V^-$ are each absolutely irreducible then I think $X^-$ is representable. In the case that $V^-$ is a line, e.g. $k_1 = 0$, then one can proceed as in the proof that ordinary deformations of an ordinary representation are *relatively* representable (regardless now of the irreducibility of $V/V^-$).

Is there a reference which discusses the representability of this functor and the relatively representability of $X^- \rightarrow X$?

The content of the statement is a descent one

If one has a deformation $V_A \in X(A)$ and $A \subset A'$ such that $V_A \otimes_A A'$ is in $X^-(A')$, then is $V_A \in X^-(A)$?

Again, I essentially know how to do this when $V^-$ is a line.