The open-closed holography partition-function conjecture

Let $\mathcal{P}_{\mathcal{C},\lambda,h,s}=\{\mathcal{C},\lambda,h,s\}$ be a holography package consisting of a smooth cyclic $A_\infty$-category $\mathcal{C}$, an object $\lambda\in\mathcal{C}$, a trivialization $(d_{hoch}+uB)h=ch(\lambda)$, and a splitting $s$. Let $Z^{c,br}(\mathcal{P}_{\mathcal{C},\lambda,h,s})$ and $Z^o(\mathcal{P}_{\mathcal{C},\lambda,h,s})$ be the backreacted closed and large $N$ open string field theory partition functions, respectively. Write $\iota_c$ and $\iota_o$ for the natural inclusion maps into $\mathcal{F}^{c}(\mathcal{C})^{Triv}\otimes\mathcal{F}^{o}(\lambda)$, and let $d_{hoch}+d+\Delta_o$ be the induced differential. Conjecture H. The partition function of the backreacted closed string field theory is homologous to that of the large $N$ open string field theory:

$$\iota_c\bigl(Z^{c,br}(\mathcal{P}_{\mathcal{C},\lambda,h,s})\bigr)\simeq \iota_o\bigl(Z^o(\mathcal{P}_{\mathcal{C},\lambda,h,s})\bigr),$$

as elements of

$$H_*\bigl(\mathcal{F}^{c}(\mathcal{C})^{triv}\otimes\mathcal{F}^{o}(\lambda),d_{hoch}+d+\Delta_o\bigr).$$

This is the partition-function formulation of open-closed holography: the closed theory obtained by backreaction in the direction of the brane $\lambda$ should agree homologically with the large $N$ open theory. The supplied text gives no evidence that the conjecture has been proved or refuted.