The subgraph core conjecture for connected graphs

Let $G$ be a graph. For each vertex $v\in V(G)$, let $f_G(v)$ be the number of connected subgraphs of $G$ containing $v$, and let $S_c(G)$ be the set of vertices with maximum subgraph number, called the subgraph core of $G$.

Subgraph core conjecture. The subgraph core of $G$ is contained in a block of $G$.

This conjecture extends the known property that central parts of graphs lie in a block. For trees, the subgraph core is already known to consist of either one vertex or two adjacent vertices; the asserted containment in a block is not resolved in the supplied source.