A gap conjecture for Sylow numbers

Let $p$ be an odd prime, and let $G$ be a finite group with subgroup $H$. For a finite group $X$, write $u_p(X)$ for the number of Sylow $p$-subgroups of $X$. Gap conjecture for Sylow numbers. There \exists a rational number $f(p)<1$ such that, whenever $u_p(H)< u_p(G)$, one has

$$\nu_p(H)\leq f(p)\cdot\nu_p(G).$$

This conjecture asks for a uniform gap, depending only on the odd prime $p$, between unequal Sylow numbers of a finite group and its subgroup. The surrounding discussion notes that equality in the general inequality $\nu_p(H)\leq\nu_p(G)$ can occur, while the proposed statement concerns the case of strict inequality; no resolution is given in the source.