Zero forcing and path number under vertex sums

Let $G$ and $H$ be graphs with an identified vertex $v$. Write G\,\makebox[0ex][l]{+} \raisebox{-.75ex}{\makebox[2ex]{_{v}}}\,H for their vertex sum at $v$. Suppose that $Z(G)=P(G)$ and $Z(H)=P(H)$. Vertex-sum conjecture. Then

Z(G\,\makebox[0ex][l]{$+$} \raisebox{-.75ex}{\makebox[2ex]{$_{v}$}}\,H)=P(G\,\makebox[0ex][l]{$+$} \raisebox{-.75ex}{\makebox[2ex]{$_{v}$}}\,H).

This proposed closure property would provide a larger family of graphs for which the zero forcing number equals the path number, extending the known examples of block-cycle graphs. The source presents it as a problem for further study; its resolution is not given here.