Hypercover descent conjecture for locally free sheaves on soft ringed spaces

Let $(X,\mathcal{A})$ be a compact ringed space with soft structure sheaf $\mathcal{A}$. Let $U_{\bullet}$ be a hypercover of $X$, and let $\mathfrak{L}(X)$ denote the dg-category of locally free sheaves on $(X,\mathcal{A})$, with $\mathfrak{L}(U_{\bullet})$ the resulting cosimplicial dg-category. Hypercover descent conjecture. The natural functor

$$\mathfrak{L}(X) \rightarrow \operatorname{holim}_{U_{\bullet}} \mathfrak{L}(U_{\bullet})$$

is a quasi-equivalence. This is proposed as a generalization from finite good covers to hypercovers of the descent result for locally free sheaves and twisted complexes; its resolution is not given in the source.