Eventual upper-bound conjecture for odd generalized Davenport constants

Let $G$ be a finite non-cyclic abelian group, and let ${\sf D}(G)$ denote its Davenport constant. For each $k\in\mathbb N$, let $\rho_{2k+1}(G)$ be the corresponding odd generalized elasticity invariant.

Eventual upper-bound conjecture. If ${\sf D}(G)\ge 4$, then there exists $k_0\in\mathbb N$ such that

$$\rho_{2k+1}(G)=k{\sf D}(G)+\left\lfloor\frac{{\sf D}(G)}{2}\right\rfloor$$

for each $k\ge k_0$.

The displayed value is the upper bound for $\rho_{2k+1}(G)$ established in the surrounding results. The lower bound is attained for cyclic groups, while this conjecture predicts eventual attainment of the upper bound for every finite non-cyclic abelian group with Davenport constant at least $4$.