The gap conjecture for products of faithful characters in finite p-groups

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, and let $\chi, \psi \in \operatorname{Irr}(G)$ be faithful characters. Let $\eta(\chi,\psi)$ denote the number of distinct irreducible constituents of $\chi\psi$. Gap conjecture. If

$$2\eta(\chi,\psi)-1>p,$$

then

$$\eta(\chi,\psi)\geq p.$$

This conjecture asks whether there are further gaps among the possible values of $\eta(\chi,\psi)$ for products of faithful irreducible characters of finite $p$-groups, beyond the established exclusion of $2\leq 2\eta(\chi,\psi)\leq p$.