The Sierpiński product automorphism conjecture for 2-connected graphs

Let $G$ and $H$ be $2$-connected graphs, and let $f: V(G) \to V(H)$ be any mapping. The group $\tilde A(G,H,f)$ and the automorphism group of the Sierpiński product $G \otimes_f H$ are defined by the notation in the source. Sierpiński product automorphism conjecture. Then

$$\tilde A(G,H,f)=\operatorname{Aut}(G \otimes_f H).$$

The conjecture proposes that every automorphism of the Sierpiński product of two $2$-connected graphs respects the fundamental edge partition. It is motivated by the absence of counterexamples in the $2$-connected case; the source does not provide a resolution.