Strongly minimal covers by independent sets

Let $G$ be a graph. An independent set is a set of vertices containing no edge of $G$, and a cover of the vertex set is a family of such sets whose union is the vertex set. Strong minimality is the dual notion to strong maximality: a cover is strongly minimal when it is no larger, in the corresponding cardinal comparison, than any alternative cover. Strongly minimal cover conjecture. Every graph has a strongly minimal cover of its vertex set by independent sets. A positive answer would yield further consequences for posets of bounded width, while the existence of such covers remains open in the stated generality.