Minimum order of an extremal bipartite graph

Let $a(G)$ be the annihilation number, $\alpha(G)$ the independence number, and $\mu(G)$ the matching number of a bipartite graph $G$. The graph is called extremal when equality holds in the bipartite bound

$$a(G)-\alpha(G)=2+\mu(G)-2\sqrt{1+\mu(G)}.$$

Minimum-order conjecture. The minimum number of vertices of an extremal bipartite graph is $16$.

The claim identifies the smallest order at which equality in the paper's sharp bipartite inequality can occur. Its resolution is not indicated in the supplied material, so it remains open in this record.