Dense 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 $\operatorname{val}_{\mathrm{IP}}$ for the integer-program value. Let $C_2$ be a constant. Dense conjecture. For $mp \gg \log n$,

$$\frac{\operatorname{val}_{\mathrm{GR}}}{\operatorname{val}_{\mathrm{IP}}} \to C_2,$$

where $1\leq C_2<1.5$. This conjecture concerns the dense regime of the random hitting set problem and predicts a constant asymptotic ratio between the greedy and integer-program solutions. It is presented as a numerical conjecture, with no resolution supplied in the source.