Triangular projection on $\boldsymbol{S}_p,~0<p<1$, as $p$ approaches $1$

This is a continuation of our recent paper. We continue studying properties of the triangular projection ${\mathscr P}_n$ on the space of $n\times n$ matrices. We establish sharp estimates for the $p$-norms of ${\mathscr P}_n$ as an operator on the Schatten–von Neumann class $\boldsymbol{S}_p$ for $0<p<1$. Our estimates are uniform in $n$ and $p$ as soon as $p$ is separated away from $0$. The main result of the paper shows that for $p\in(0,1)$, the $p$-norms of ${\mathscr P}_n$ on ${\mathscr P}_n$ behave as $n\to\infty$ and $p\to1$ as $n^{1/p-1}\min\big\{(1-p)^{-1},\log n\big\}$.