Categorical DT wall-crossing conjecture for stable-pair moduli

Fix $\sigma=iH$ for an ample divisor $H$, $v=(\beta,n)\in N_{\leq 1}(S)$, and a positive rational wall parameter $t_0\in W\cap\mathbb{Q}_{>0}$. For $t_{\pm}=t_0\pm\varepsilon$ with $0<\varepsilon\ll1$, let $P_n(X,\beta)_{t_{\pm}}$ be the stable-pair moduli spaces in the adjacent chambers. Categorical DT wall-crossing conjecture. There exists a fully faithful functor

$$\mathcal{DT}^{\mathbb{C}^{\ast}}(P_n(X,\beta)_{t_-})\hookrightarrow\mathcal{DT}^{\mathbb{C}^{\ast}}(P_n(X,\beta)_{t_+}).$$

This is the categorical prediction for the d-critical flip at the wall $t_0$; the source notes that the corresponding conjecture is proved in the special case where $S\to\mathbb{C}^2$ is the blow-up at the origin, but gives no general resolution.