The uniform hypergraph bound for the vertex-stable-set invariant

Let ${\cal H}=(V,{cal E})$ be a hypergraph with $|V|=n$, no isolated vertices, and every edge $h\in{cal E}$ of size $r$. Let $\sigma_v({\cal H})$ and $\alpha({\cal H})$ be the stated vertex-stable-set invariants. The uniform hypergraph conjecture. If

$$\sigma_v({\cal H})=\alpha({\cal H}),$$

then

$$rsigma_v({\cal H})\leq n.$$

This conjecture was attributed in the source to an earlier thesis; no resolution is given in the supplied text.