The alternative-Hopf-structure conjecture for monomial modules

Let $G=\mathbb{Z}/2^r\mathbb{Z}\times\mathbb{Z}/2^s\mathbb{Z}$, and let $\Bbbk\alpha(r,s)$ be the Hopf algebra with the alternative comultiplication used to define the tensor product and dual. Let $\otimes$ and $(-)^*$ denote these operations, and let $\otimes'$ and $(-)^\vee$ denote the operations defined using the standard Hopf algebra structure on $\Bbbk G$. For an odd-dimensional indecomposable monomial representation $V$, the alternative-Hopf-structure conjecture. There are isomorphisms of $G$-representations

$$V\otimes V^*\cong V\otimes' V^\vee,\qquad V^{\otimes n}\cong V^{\otimes' n}.$$

The claim is based on computational evidence for the monomial representations studied in the paper and remains conjectural.