The edge domination conjecture for connected regular graphs

Let $G$ be a finite, simple, connected, $\Delta$-regular graph of order $n$, where $\Delta\geq 3$. The edge domination conjecture asserts that

$$\gamma_e(G)\leq \frac{2\Delta-1}{4\Delta}n+\frac{1}{2}.$$

Equality holds if and only if $G$ has a spanning subgraph that is the union of an odd number of copies of $K_{\Delta,\Delta}-e$. This conjecture would give a best-possible upper bound on the minimum size of a maximal matching in connected regular graphs, with an explicit characterization of the equality cases.