  Let $G$ be a $2$-step stratified group of topological dimension $d$ and
homogeneous dimension $Q$. Let $L$ be a homogeneous sub-Laplacian on $G$. By a
theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$
is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$
whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of
order $s > Q/2$. It is known that, for several $2$-step groups and
sub-Laplacians, the threshold $Q/2$ in the smoothness condition is not sharp
and in many cases it is possible to push it down to $d/2$. Here we show that,
for all $2$-step groups and sub-Laplacians, the sharp threshold is strictly
less than $Q/2$, but not less than $d/2$.
