TxGraffiti's annihilation-number bound for connected graphs

Let $G$ be a nontrivial connected graph. Write $4\alpha(G)4$ for its independence number, $4a(G)4$ for its annihilation number, $4R(G)4$ for its residue, and $4\Delta(G)4$ for its maximum degree.

TxGraffiti's conjecture. One has

$$\alpha(G) \ge \frac{a(G)+R(G)}{\Delta(G)},$$

and this bound is sharp.

The conjecture is open in general; it is established for regular bipartite and cubic Kőnig–Egerváry graphs. The bound is motivated by automated graph conjecturing and is sharp, for example, for the complete graph on four vertices.