Nullity equality conjecture for singular io-decomposable Riordan graphs

Let ${\cal A}_n$ be the adjacency matrix of a singular io-decomposable Riordan graph $G_n(g,zg)$ of the form specified in the preceding structural theorem, and let $B$ be the associated off-diagonal block in the block decomposition of ${\cal A}_n$. Write $\eta(H)$ for the nullity of a graph or its adjacency matrix. Nullity equality conjecture. Then

$$\eta(G_n)=\eta(B).$$

This is presented as an equivalent version of an earlier conjecture, using the rank calculation established immediately before it; the supplied text gives no resolution.