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, $\operatorname{val}_{\mathrm{IP}}$ for the integer-program value, and $\operatorname{val}_{\mathrm{LP}}$ for the linear-programming relaxation. Let $C_1$ be a constant. Sparse conjecture. For $1 \lesssim mp \ll \log n$,

$$\frac{\operatorname{val}_{\mathrm{GR}}}{\operatorname{val}_{\mathrm{IP}}} \to 1, \qquad \frac{\operatorname{val}_{\mathrm{IP}}}{\operatorname{val}_{\mathrm{LP}}} \to C_1,$$

where $1<C_1<1.5$. This conjecture describes the intermediate-sparsity regime and predicts both asymptotic optimality of the greedy algorithm relative to the integer program and a bounded nontrivial integrality gap. The source formulates it from numerical experiments and gives no resolution.