Equality of relative Davenport constants after removing a smaller coset

Let $G$ be a finite abelian group and $H_1<H_2\leq G$. For some $g\in G$, write

$$\alpha=g+H_1,\qquad \beta=g+H_2,$$

so that $\alpha\subsetneq\beta$.

Relative Davenport equality conjecture. One has

$$d_\beta(G)=d_{\beta\backslash\alpha}(G).$$

Equivalently, there exists a $\beta$-sequence $\{g_1,\dots,g_d\}$ of length $d=d_\beta(G)$ with no zero-sum subsequence in $G$ whose sum does not lie in the proper subset $\alpha$. The claim is presented as a conjectural extension of the preceding cyclic-group corollary to arbitrary finite abelian groups.