Let $G = G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$, and let $\overline{D}$ be a residual pseudorepresentation of $\mathbb{Z}_p[G]$ over $\mathbb{F}_p$.
Denote by $\text{Rep}^d_{\overline{D}}$ the functor which associates to a rigid analytic space $X/\text{Spm}(\mathbb{Q}_p)$ the category of locally free rank-$d$ $\mathcal{O}_X$-modules, equipped with a continuous linear action of $G$, and having residual pseudorepresentation $\overline{D}$.
A famous theorem of Wang-Erickson (see here) says that the functor $\text{Rep}^d_{\overline{D}}$ is representable as a formal algebraic stack over some universal deformation ring associated to the residual pseudorepresentation, $\overline{D}$.
I wonder if it is possible that the functor $\text{Rep}^d_{\overline{D}}$ is representable by an algebraic stack instead. Is there some reason why this shouldn't be the case? How does one go about proving such a statement? I am particularly curious about the case $d=2$, in case it should be any easier.
Finally, are there any reading references you might recommend to me so I could get a more structured exposition into this topic.
Thanks in advance!