Tree-extension conjecture for Schubert-cell decompositions of quiver Grassmannians

Let $T$ be a tree extension of $S$, and let $M$ be a $T$-module such that $M_\alpha$ is an isomorphism for all arrows $\alpha$ in $T-S$. Let $M_S$ be the restriction of $M$ to $S$, and let ${\mathcal B}$ be an ordered basis of $M$ whose restriction ${\mathcal B}_S={\mathcal B}\cap M_S$ is ordered above it. Let $\beta\subset{\mathcal B}$ be of type ${\underline e}$ and let $\beta_S=\beta\cap M_S$ be of type ${\underline e}_S$. Tree-extension conjecture. If $\overline{C_{\beta_S}^{M_S}}$ decomposes into Schubert cells, then $\overline{C_\beta^M}$ also decomposes into Schubert cells. Moreover, if

$$\overline{C_{\beta_S}^{M_S}}=\coprod_{\gamma_S\preceq\beta_S\text{ of type }{\underline e}_S} C_{\beta'_S}^{M_S},$$

then

$$\overline{C_\beta^M}=\coprod_{\beta'\preceq\beta\text{ of type }{\underline e}} C_{\beta'}^M.$$

In particular, if $\operatorname{Gr}_{{\underline e}_S}(M_S)=\coprod C_{\beta_S}^{M_S}$ is a regular decomposition, then so is $\operatorname{Gr}_{{\underline e}}(M)=\coprod C_\beta^M$. The statement proposes that regular Schubert decompositions are preserved under tree extensions with isomorphic maps on the added arrows; the source gives no resolution.