  We define a strengthening of the Haagerup-Kraus approximation property by
means of the subalgebras of Herz-Schur multipliers $M_d(G)$ ($d\geq 2$)
introduced by Pisier. We show that unitarisable groups satisfying this property
for all $d\geq 2$ are amenable. Moreover, we show that groups acting properly
on finite-dimensional CAT(0) cube complexes satisfy $M_d$-AP for all $d\geq 2$.
We also give examples of non-weakly amenable groups satisfying $M_d$-AP for all
$d\geq 2$.
