Bollobás–Thomason threshold conjecture for binomial random multigraphs

Let $k\ge 3$ be fixed, and let $\mathcal{P}$ be a non-trivial monotone increasing (multi)graph property. A threshold in $\mathbb{G}(n,p\,; \,\binom{[n]}{k})$ is a sequence $\{c_n^{\ast}\}_n$ such that

$$\lim_{n\to\infty} \mathbb{P}(\mathbb{G}(n,p_n\,; \,\binom{[n]}{k}) \in\mathcal{P}) = \begin{cases} 0,&\text{if } p_n \ll c_n^{\ast} \\ 1,&\text{if } p_n \gg c_n^{\ast}. \end{cases}$$

Bollobás–Thomason threshold conjecture. The property $\mathcal{P}$ has a threshold in $\mathbb{G}(n,p\,; \,\binom{[n]}{k})$.

This conjecture proposes that the Bollobás–Thomason theorem for monotone increasing graph properties extends to these binomial random multigraphs, where each $k$-set in $\binom{[n]}{k}$ provides the underlying multigraph structure. The source gives no resolution evidence, so the conjecture is recorded as open.