It has been proved by Toën and Lunts-Schnürer that the dg category $\mathrm{L}_{qcoh}(X\times Y)$ of quasi-coherent sheaves over the product of two quasi-compact, quasi separated (and flat over a commutative ring $k$) schemes is quasi-equivalent to the dg-category of continuous dg functors $\mathbb{R}\underline{\mathrm{Hom}}_c(\mathrm{L}_{qcoh}(X),\mathrm{L}_{qcoh}(Y))$, thus providing a kernel for every triangulated functor between the derived categories of $X$ and $Y$ which is eligible for being of Fourier-Mukai type (i.e. commuting with direct sums). This equivalence also descends to the perfect dg subcategories and the internal (derived) Hom for dg categories above is defined by Toën in his paper https://arxiv.org/abs/math/0408337 by means of some dg bi-modules over the two dg categories.

What is known about whether this equivalence still holds when we consider the dg categories of complexes of (perfect or not) twisted sheaves? The paper https://arxiv.org/abs/1002.2599 by Toën proves that that the twisted categories have a generator, is that sufficient to claim some result à la Bondal-Van den Bergh saying that the box product of the two generators is the generator of the dg category of the product? To summarize in one line: is it plausible to have a result like $$ \mathbb{R}\underline{\mathrm{Hom}}_c(\mathrm{L}_{qcoh}(X,\alpha),\mathrm{L}_{qcoh}(Y,\beta)) \simeq \mathrm{L}_{qcoh}(X\times Y, \alpha^{-1}\boxtimes \beta) $$ eventually for perfect complexes? and if it is so plausible that someone already wrote it, where could I read it?

Thank you!