Subquotient conjecture for quantum groups under edge contractions

Let $\Gamma$ be a graph, let $e$ be an edge of $\Gamma$, and let $\Gamma/e$ denote the graph obtained by contracting $e$. Write $\mathbf U_{\Gamma/e}$ and $\mathbf U_{\Gamma}$ for the associated quantum groups, and let the edge contraction induce the explicit embedding

$$\mathbf U_{\Gamma/e}\hookrightarrow \mathbf U_{\Gamma}.$$

Subquotient conjecture. The explicit embedding for quantum groups in Theorem~ is a section of a subquotient. This conjecture is motivated by the corresponding result for Hall algebras and their positive and negative halves; the precise subquotient structure for the full quantum groups is proposed but not established in the supplied text.