  Grothendieck's theorem asserts that every continuous linear operator from
$\ell_{1}$ to $\ell_{2}$ is absolutely $\left( 1;1\right) $-summing. In this
note we prove that the optimal constant $g_{m}$ so that every continuous
$m$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is
absolutely $\left( g_{m};1\right) $-summing is $\frac{2}{m+1}$. We also show
that if $g_{m}<\frac{2}{m+1}$ there is $\mathfrak{c}$ dimensional linear space
composed by continuous non absolutely $\left( g_{m};1\right) $-summing
$m$-linear operators from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}.$
In particular, our result solves (in the positive) a conjecture posed by A.T.
Bernardino in 2011.
