Spanning-subtree probability conjecture for the complete graph

Let $G$ be a graph with $n$ vertices, let $s_k(G)$ be the number of subtrees of $G$ with $k$ vertices, and define the spanning-subtree probability

$$p(G)=\frac{s_n(G)}{\sum_{k=1}^n s_k(G)}.$$

Spanning-subtree probability conjecture. For every graph $G$ with $n$ vertices,

$$p(G)\le p(K_n).$$

The source states that this less restrictive conjecture is an open problem and that it would support the extremality of the complete graph.