  In [18] we have shown that, for $p_{1},p_{2}\in(2,\infty]$, the constants of
Bennett's inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1}
}\times\ell_{p_{2}}^{n_{2}}$ are asymptotically bounded by $1$. In the present
paper we use a different approximation technique to investigate the remaining
cases $p_{1},p_{2}\in\lbrack1,\infty].$ This new approach also provides a
stronger asymptotic control, in the sense that the constants are "uniformly"
asymptotically bounded by $1$, with no dependence on $p_{1},p_{2}.$
