On a norm compression inequality for 2×N2×N partitioned block matrices

We conjecture the following so-called norm compression inequality for 2×N2×N partitioned block matrices and the Schatten p  -norms: for p⩾2p⩾2,A1A2⋯ANB1B2⋯BNp⩽‖A1‖p‖A2‖p⋯‖AN‖p‖B1‖p‖B2‖p⋯‖BN‖ppwhile for 1⩽p⩽21⩽p⩽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 show that the partitioning in 2×N2×N blocks is essential, by exhibiting counterexamples in the case of 3×33×3 and larger partitionings.