The semidirect-product conjecture for Sierpiński product automorphisms

Let $G$ and $H$ be connected graphs on the same vertex set, and let $f: V(G) \to V(H)$ be a bijection. The groups $\tilde A(G,H,f)$, $\bar A(G,H,f)$, and $\hat B(G,H,f)$ are the automorphism groups and subgroups defined in the source. Semidirect-product conjecture. The group $\tilde A(G,H,f)$ is a semidirect product:

$$\tilde A(G,H,f)=\bar A(G,H,f) \ltimes \hat B(G,H,f).$$

This extends the preceding semidirect-product result for the case $G=H$ and $f$ an automorphism. The source presents the assertion as believed to hold and gives no resolution.