Weakly-filtered endomorphism algebra conjecture

Let $E_0$ and $E_1$ be the objects appearing in the surgery-formula construction, and let $\operatorname{End}_{w\mathrm{Fil}}(\beta_0^{E_0}\oplus\beta_1^{E_1})$ denote the endomorphism space satisfying the weakly-filtered conditions, including morphisms from $\beta_1^{E_1}$ to $\beta_0^{E_0}$. Consider its chiral and $U$-adic topologies, and let $\mathcal{K}$ and $\mathfrak{K}$ be the corresponding algebras. Weakly-filtered endomorphism algebra conjecture. For both the chiral and $U$-adic topologies, the space $\operatorname{End}_{w\mathrm{Fil}}(\beta_0^{E_0}\oplus\beta_1^{E_1})$ is an $A_\infty$-algebra. Furthermore, it is $A_\infty$-homotopy equivalent to either $\mathcal{K}$ in the chiral case or $\mathfrak{K}$ in the $U$-adic case. This conjecture asserts that allowing the additional weakly-filtered morphisms does not change the relevant $A_\infty$-homotopy type; the source gives no evidence of a resolution, so it remains open.