Very sparse integrality-gap conjecture for random hitting set

Let $m$ be the number of subsets, let $p$ be the inclusion probability, and let $n$ be the number of elements. Write $\operatorname{val}_{\mathrm{GR}}$ for the value produced by the greedy algorithm and $\operatorname{val}_{\mathrm{LP}}$ for the linear-programming relaxation. Very sparse conjecture. For $mp \ll 1$,

$$\frac{\operatorname{val}_{\mathrm{GR}}}{\operatorname{val}_{\mathrm{LP}}} \to 1.$$

This conjecture concerns the very sparse regime of the random hitting set problem and predicts that the greedy solution is asymptotically optimal relative to the linear relaxation. The source presents it as a conjecture based on numerical experiments; its resolution is not given.