TxGraffiti's independence–matching conjecture for regular graphs

Let $G$ be a connected $r$-regular graph with $r>0$. Let $\alpha(G)$ denote the independence number of $G$, and let $\mu(G)$ denote its matching number. TxGraffiti's independence–matching conjecture. Then

$$\alpha(G)\leq\mu(G),$$

and this bound is sharp. This is a generalized form of the example conjecture produced by the filtering heuristic, illustrating how a more general graph hypothesis can be retained for the same sharp inequality.