Strong conjecture on transversals of large maximal independent sets

Let $0<c<1$ be a constant. For a graph $G$, let $\operatorname{MIS}(G)$ denote its family of maximal independent sets, and define the family of large maximal independent sets by

$$\operatorname{MIS}_{\mathrm{large}}(G)=\{I\in\operatorname{MIS}(G): |I|\geq cn\}.$$

A transversal is a set of vertices meeting every member of this family. Strong conjecture. Every graph $G$ on $n$ vertices has a transversal of size $o(n)$ for $\operatorname{MIS}_{\mathrm{large}}(G)$. The source describes this as a stronger conjecture proposed tentatively, and provides no proof or counterexample; the supplied status evidence is inconsistent with the assertion and is therefore noted for review.