Quasi-invertibility conjecture for bordered Koszul-duality bimodules

Let $\mathcal{B}$ be the algebra associated to a bordered theory, let $m(\mathcal{B})$ denote its mirror, let $\astrosun$ be the relevant product, let $\mathcal{B}\Gamma_n$ be the quotient algebra, and let $K$ denote the displayed Type DD bimodules. The notation $m(\mathcal{B})^!$, $\mathcal{B}\astrosun m(\mathcal{B})^!$, and $m(\mathcal{B}\astrosun m(\mathcal{B})^!)$ is as in the paper.

Quasi-invertibility conjecture. Either or both of the DD bimodules

$${}^{\mathcal{B}\astrosun m(\mathcal{B})^!}K^{m(\mathcal{B}\astrosun m(\mathcal{B})^!)^{op}}$$

and

$${}^{\mathcal{B}\Gamma_n}K^{m(\mathcal{B}\Gamma_n)^{op}}$$

are quasi-invertible. Hence, either or both of the algebras $\mathcal{B}\astrosun m(\mathcal{B})^!$ and $\mathcal{B}\Gamma_n$ are Koszul dual to their mirrors, $m(\mathcal{B}\astrosun m(\mathcal{B})^!)$ and $m(\mathcal{B}\Gamma_n)$, in the generalized sense.

This proposes generalized Koszul duality between each bordered algebra and its mirror, mediated by a quasi-invertible Type DD bimodule. The parser supplies no evidence that the claim has been resolved, so its status remains open.