Bollobás–Erdős–Tuza conjecture on transversals of maximum independent sets

Let $0<c<1$ be a constant. For a graph $G$, let $\alpha(G)$ be the size of a largest independent set, and let $h(G)$ be the size of a smallest set meeting every maximum independent set. Bollobás–Erdős–Tuza maximum conjecture. If $G$ is a graph on $n$ vertices with

$$\alpha(G)\geq cn,$$

then

$$h(G)=o(n).$$

The source presents this as a closely related conjecture that has received recent attention; no resolution is given.