The bound $\psi(G)\leq 2$ for finite groups of nonzero spread

Let $G$ be a finite group of nonzero spread. The invariant $\psi(G)$ is the value of $r$ at which the relations $\equiv_{\mathrm{m}}^{(r)}$ stabilize to $\equiv_{\mathrm{m}}$.

Nonzero-spread bound conjecture.

$$\psi(G)\leq 2.$$

For soluble groups of nonzero spread, this bound is known: in the two-generator case, one has $\psi(G)=2$. The conjecture removes the solubility assumption and would imply that $\equiv_{\mathrm{m}}$ and $\equiv_{\mathrm{m}}^{(2)}$ coincide.