Category-theoretic and quiver-bundle moduli stack equivalence conjecture

Let $X\in\mathrm{St}_{\mathscr{C}}$ be a stack over $\mathscr{C}$, and let $I$ be a simplicial set. Define the category-theoretic moduli stack of quiver bundles by

$$\mathcal{M}_{\mathrm{Vect}(X),I}^{\mathrm{cat}}:=\mathrm{Map}(\underline{I},\mathcal{M}_{\mathrm{Vect}(X)}).$$

Let $\mathcal{M}_{\mathrm{Vect}(X),I}$ denote the previously constructed moduli stack of quiver bundles.

Category-theoretic and quiver-bundle moduli stack equivalence conjecture. There is an equivalence of stacks

$$\mathcal{M}_{\mathrm{Vect}(X),I}^{\mathrm{cat}}\simeq\mathcal{M}_{\mathrm{Vect}(X),I}$$

for all $I$. In fact, this equivalence is a natural equivalence of functors

$$\mathrm{sSet}\longrightarrow\mathrm{St}_{\mathrm{CRing}^{\heartsuit,\mathrm{op}}}^{1}.$$

This conjecture asserts that the category-theoretic mapping-stack description agrees with the previously constructed moduli stack of quiver bundles, naturally in the simplicial indexing object. The supplied text gives no resolution status.