Conjectural derived moduli stack of rank-n vector bundles

Let $X$ be a fixed smooth projective variety and let $n$ be a positive integer. For a commutative differential graded algebra $A$, let $\mathbb{R}\underline{Vect}_{n}(X)(A)$ be the nerve of the category of flat dg-$\mathcal{O}_{X}\otimes A$-modules that are vector bundles of rank $n$, with equivalences given by stalkwise quasi-isomorphisms. Write $\mathbb{R}\underline{Vect}_{n}(X)$ for the resulting derived moduli stack, let $\underline{Vect}_{n}(X)$ denote the Artin stack of rank-$n$ vector bundles on $X$, and let $E$ be a vector bundle on $X$. The conjecture. The $D$-pre-stack $\mathbb{R}\underline{Vect}_{n}(X)$ is a strongly geometric, fp-smooth $D$-stack; there is a natural isomorphism in $\mathrm{Ho}(D\text{-}Aff^{\sim})$

$$\mathbb{R}\underline{Vect}_{n}(X)\simeq\mathbb{R}\mathcal{HOM}(X,iBGl_{n});$$

one has an equivalence

$$h^{0}\mathbb{R}\underline{Vect}_{n}(X)\simeq\underline{Vect}_{n}(X);$$

and the tangent $D$-stack at $E$ is the complex

$$C^{*}(X_{Zar},\underline{End}(E))[1].$$

These assertions identify the derived moduli object with the derived mapping stack into the classifying object of $GL_n$, recover the classical moduli stack after truncation, and describe its tangent complex. The source presents the result as a conjecture because the authors had not checked all details, although they expected a proof by adapting constructions of Toën and Vaquié.