Audenaert–Kittenah Schatten norm compression conjecture

Let $T$colon $obreak\bigoplus_{j=1}^N\bigl( obreak\bigoplus\bigr)$ be the partitioned operator

$$T=\begin{pmatrix} A_1&A_2&\cdots&A_N\\ B_1&B_2&\cdots&B_N \end{pmatrix}.$$

For its Schatten $p$-norm compression, define

$$C_p(T)=\begin{pmatrix} \|A_1\|_p&\|A_2\|_p&\cdots&\|A_N\|_p\\ \|B_1\|_p&\|B_2\|_p&\cdots&\|B_N\|_p \end{pmatrix}.$$

Audenaert–Kittenah conjecture. For $1\le p\le 2$,

$$\|T\|_p\ge \|C_p(T)\|_p.$$

For $2\le p\le\infty$, the inequality is reversed. The conjecture generalizes the cited two-row Schatten inequalities; the source notes that the positive case for $N=2$ is known, while the full assertion is presented as conjectural.