Categorical Donaldson–Thomas wall-crossing conjecture for PT stable-pair moduli

Let $S$ be a smooth projective surface over $\mathbb{C}$, let $X=\operatorname{Tot}_S(\omega_S)$, and let $P_n^t(X,\beta)$ denote the moduli space of $\mu_t^{\dagger}$-semistable D0-D2-D6 objects of numerical class $(\beta,n)$, with $t_-$ and $t_+$ lying on the two sides of a wall in the wall-crossing diagram. Write $\mathcal{DT}^{\mathbb{C}^{\ast}}(P_n^t(X,\beta))$ for the associated $\mathbb{C}^{\ast}$-equivariant DT category. Categorical DT wall-crossing conjecture. There exists a fully faithful functor

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

This is a categorical analogue of the D/K-equivalence principle for birational wall crossing and predicts that the DT category on the $t_-$ side embeds into that on the $t_+$ side. The supplied text does not state whether the conjecture has been proved or disproved.