  The vortex contribution to the $k$-string tensions is computed for SU($N$)
gauge theories. We deduce the surface densities needed to reproduce the sine
scaling and the Casimir scaling formulae, recently obtained from numerical
simulations on the lattice. We find that such densities need to grow linearly
in $N$, which in turn suggests that the vortex scenario can hardly reproduce
the physics of confinement at large $N$.
