The dimension-zero–dimension-one commutation conjecture for surface Hall algebras

Let $S$ be the surface in the preceding setup, let $C_i$ be one of the distinguished curves, and let $[\mathfrak Y_{i,(d)}]$ and $[\mathfrak Z_{i,\ell}]$ denote the corresponding dimension-zero and dimension-one Hall classes, with $\ell\in\mathbb Z$ and $d\in\mathbb N$. The dimension-zero–dimension-one commutation conjecture. Without assuming $\widehat S_{C_i}\simeq(\widehat{T_i}_{C_i})$, one still has

$$\left[[\mathfrak Y_{i,(d)}],[\mathfrak Z_{i,\ell}]\right]=(C_i\cdot C_i)[\mathfrak Z_{i,\ell+d}].$$

The claim extends the preceding commutation relation beyond the formal-neighborhood hypothesis. The supplied text gives no proof or resolution, so its status is open.