The nullity conjecture for connected block graphs with blocks of order at least 3

Let $G$ be a connected block graph, meaning a graph in which every biconnected component is a clique. The nullity of $G$ is the dimension of the kernel of its adjacency matrix.

Nullity conjecture. If each block of $G$ has order at least $3$, then the nullity of $G$ is at most $1$.

This conjecture concerns the adjacency-matrix nullity of $K_2$-forbidden block graphs. Its resolution is not indicated in the source.