Categorical DT wall-crossing conjecture for MNOP and stable-pair theories

Fix $(\beta,n)$ and choose stability parameters $t_1>t_2>\cdots>t_N>0$ away from the finitely many walls. Let $I_n(X,\beta)$ be the MNOP moduli space, $P_n(X,\beta)$ the stable-pair moduli space, and $P_n(X,\beta)_{t_i}$ the moduli spaces for the corresponding stability parameters. Write $\mathcal{DT}^{\mathbb{C}^{\ast}}(-)$ for the associated $\mathbb{C}^{\ast}$-equivariant categorical DT theories. Categorical DT wall-crossing conjecture. There exist fully faithful functors

$$\mathcal{DT}^{\mathbb{C}^{\ast}}(P_n(X,\beta)_{t_N})\hookrightarrow\cdots\hookrightarrow\mathcal{DT}^{\mathbb{C}^{\ast}}(P_n(X,\beta)_{t_1})\hookrightarrow\mathcal{DT}^{\mathbb{C}^{\ast}}(P_n(X,\beta))\hookrightarrow\mathcal{DT}^{\mathbb{C}^{\ast}}(I_n(X,\beta)).$$

This packages the expected categorical behavior of the d-critical minimal model program for MNOP/PT wall crossing; the source gives no general resolution.