Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)

This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||mathcal{E}(rho)-mathcal{E}(sigma) ||leq k ||rho-sigma||$, for $0leq k <1$.

The requirement for perfect error correction of a subspace $K$ is that there exists a channel $S$ such that $S$ is the inverse of the restriction of $mathcal{E}$ to the subspace $K$.

The proof of impossibility uses the fact that this would require $||Smathcal{E}(|uranglelangle u|)-Smathcal{E}(|vranglelangle v|)|| = |||uranglelangle u|-|vranglelangle v|||$, for some basis vectors $u,v$, which would contradict strict contractivity.

My confusion is concerning how this contradiction argument doesn’t seem take into consideration the fact that we should restrict to the subspace $K$. In other words, if $P$ is the projector onto the subspace $K$, is it generally true that if $mathcal{E}$ is strictly contractive, then
$||P(mathcal{E}(rho))-P(mathcal{E}(sigma)) ||<||P(rho)-P(sigma)||$?

Thank you in advance.