  From a theorem of Christ and Mauceri and Meda it follows that, for a
homogeneous sublaplacian $L$ on a $2$-step stratified group $G$ with Lie
algebra $\mathfrak{g}$, an operator of the form $F(L)$ is of weak type $(1,1)$
and bounded on $L^p(G)$ for $1 < p < \infty$ if the spectral multiplier $F$
satisfies a scale-invariant smoothness condition of order $s > Q/2$, where $Q =
\dim \mathfrak{g} + \dim[\mathfrak{g},\mathfrak{g}]$ is the homogeneous
dimension of $G$. Here we show that the condition can be pushed down to $s >
d/2$, where $d = \dim \mathfrak{g}$ is the topological dimension of $G$,
provided that $d \leq 7$ or $\dim [\mathfrak{g},\mathfrak{g}] \leq 2$.
