Conjecture on the semicanonical functions for semistable diagonals

Let $I'=I\setminus\{p\}$ index the irreducible components $Y_k$ of the exceptional variety $\mathfrak{L}=\mathfrak{L}_m$, and let $\widetilde E_k(m)$ be the characteristic function of $Y_k$. For $\dim V=m\delta$, let $\Lambda^{ssd}_V$ be the semistable diagonal, and let $\hat{E}^{*}_k(m)$ be the commutator function defined from $E^{*}_{\alpha_k}$ and $E^{*}_{\beta_k}$, where $\alpha_k=k$ and $\beta_k=m\delta-\alpha_k$. Semicanonical-function conjecture. Up to signs, the following identities hold: $E^{*}_{\alpha}|_{\Lambda^{s}_V}=\widetilde E_{\alpha}$ for $\dim V=\alpha\in R^{\operatorname{re}}_{+}$; $\hat{E}^{*}_k(1)|_{\Lambda^{s}_V}=\widetilde E_k(1)$ for $\dim V=\delta$; and $\hat{E}^{*}_k(m)|_{\Lambda^{ssd}_V}=\widetilde E_k(m)$ for $\dim V=m\delta$. In addition, if $m>1$ and $x',x”\in\Lambda^{ssd}_V$ lie in the same $S$-equivalence class with both $\hat{E}^{*}_k(m)(x')$ and $\hat{E}^{*}_k(m)(x”)$ nonzero, then these two values are equal, so that $\hat{E}^{*}_k(m)$ descends to a function on $\Lambda^{ssd}_V/S$-equivalence by taking the common nonzero value when it exists and $0$ otherwise. The conjecture identifies the commutator functions with characteristic functions of the irreducible components and is intended to establish the relevant semicanonical-basis elements on the semistable diagonal; its resolution is not given in the source.