The induced-matching bound for connected graphs beyond $C_5^2$ and $K_{3,3}^+$

Let $G$ be a connected graph with maximum degree $\Delta\geq 3$. Let $C_5^2$ be the graph obtained from a $5$-cycle by replacing its vertices with independent sets of order $2$, and let $K_{3,3}^+$ be the graph obtained from $K_{3,3}$ by subdividing exactly one edge once. Write $n(G)$ for the number of vertices, and let $u_s(G)$ denote the maximum size of an induced matching in $G$. The induced-matching bound conjecture. If $G\notin\{C_5^2,K_{3,3}^+\}$, then

$$\nu_s(G) \geq \frac{1}{\left(\left\lceil\frac{\Delta}{2}\right\rceil+1\right)\left(\left\lfloor\frac{\Delta}{2}\right\rfloor+1\right)}n(G).$$

The conjecture is motivated by a theorem giving this bound for sufficiently large maximum degree and by the exceptional graphs $C_5^2$ and $K_{3,3}^+$, which show that the corresponding theorem cannot extend unchanged to maximum degree three or four. Its validity for all connected graphs satisfying the stated hypotheses remains open.