It is argued on the basis of the BCS theory that the zero-$T$ penetration depth satisfies ${\lambda }^{-2}\left(0\right)\propto \sigma T_c$ ($\sigma$ is the normal state dc conductivity) not only in the extreme dirty limit ${\xi }_0/\ell \gg 1$, but in a broad range of scattering parameters down to ${\xi }_0/\ell \sim 1$ (${\xi }_0$ is the zero-$T$ BCS coherence length and $\ell$ is the mean free path). Hence, the scaling ${\lambda }^{-2}\left(0\right)\propto \sigma T_c$, suggested as a new universal property of superconductors [Dordevic, Basov, and Homes, Sci. Rep. 3, 1713 (2013)], finds a natural explanation within the BCS theory.