Product-one subsequence conjecture for the ordered Davenport constant

For a finite group $G$, let $\mathsf{D}(G)$ be the least positive integer such that every sequence of that length over $G$ has a nonempty product-one subsequence, and let $\mathsf{E}(G)$ be the least integer $k$ such that every sequence of length $k$ over $G$ has a product-one ordered subsequence of length $|G|$. The ordered Davenport constant conjecture. One has

$$\mathsf{E}(G)=\mathsf{D}(G)+|G|-1.$$

For finite abelian groups this equality is known, while the source establishes the lower bound $\mathsf{E}(G)\geq\mathsf{D}(G)+|G|-1$ for finite non-abelian groups. The conjecture asks whether equality holds for every finite group.