  In 2003, H\'{e}thelyi and K\"{u}lshammer proposed that if $G$ is a finite
group and $p$ is a prime dividing the group order, then $k(G)\geq 2\sqrt{p-1}$,
and they proved this conjecture for solvable $G$ and showed that it is sharp
for those primes $p$ for which $\sqrt{p-1}$ is an integer. This initiated a
flurry of activity, leading to many generalizations and variations of the
result; in particular, today the conjecture is known to be true for all finite
groups. In this note, we put forward a natural new and stronger conjecture,
which is sharp for all primes $p$, and we prove it for solvable groups, and
when $p$ is large, also for arbitrary groups.
