Almost-spanning subtree ratio conjecture for the complete graph

Let $G$ be a graph of order $n$. For $1\le k\le n$, let $s_k(G)$ denote the number of subtrees of $G$ with $k$ vertices; in particular, $s_n(G)$ counts spanning trees.

Almost-spanning subtree ratio conjecture. For every graph $G$ of order $n$,

$$s_{n-1}(G)\ge \frac{s_{n-1}(K_n)}{s_n(K_n)}s_n(G).$$

This is proposed as a reduction toward proving that the complete graph has greatest mean subtree order. Its status is not resolved in the supplied text.