  We conjecture the following so-called norm compression inequality for
$2\times N$ partitioned block matrices and the Schatten $p$-norms: for $p\ge
2$, $$ ||({array}{cccc} A_1 & A_2 & ... & A_N B_1 & B_2 & >... & B_N
{array})||_p
  \le ||({array}{cccc} ||A_1||_p & ||A_2||_p & ... & ||A_N||_p \ ||B_1||_p &
||B_2||_p & ... & ||B_N||_p {array})||_p $$ while for $1\le p\le 2$ the
ordering of the inequality is reversed. This inequality includes Hanner's
inequality for matrices as a special case. We prove several special cases of
this inequality and give examples for $3\times 3$ and larger partitionings
where it does not hold.
