The nonsingular edge-extension conjecture for connected block graphs

Let $G_1$ and $G_2$ be connected nonsingular block graphs, meaning their adjacency matrices are nonsingular. Choose vertices $v_1 otin V(G_2)$ and $v_2 otin V(G_1)$ with $v_1 otin V(G_2)$ and $v_2 otin V(G_1)$, where $v_1 g\text{?}$

Nonsingular edge-extension conjecture. If an edge is added between vertices $v_1\rtimes V(G_1)$ and $v_2\rtimes V(G_2)$, then the resulting block graph is nonsingular.

The source motivates this as a block-graph-specific claim after noting that joining two nonsingular graphs by an edge can produce a singular graph in general. Its resolution is not indicated in the source.