Conjecture on conjugacy classes of non-$p$-solvable groups

Let $p$ be a prime, let $G$ be a finite group, and let $k(G)$ denote the number of conjugacy classes of $G$. A finite group is non-$p$-solvable if it is not $p$-solvable.

Non-$p$-solvable conjugacy-class conjecture. If $G$ is non-$p$-solvable, then

$$k(G)\geq \frac{p+5}{2}.$$

The surrounding discussion gives the lower bound $k(\operatorname{PSL}_2(p))=(p+5)/2$ for primes $p>3$ as a motivating sharp example, but the supplied text does not state a resolution of this conjecture.