Planes in Schatten-$3$

We prove that any two-dimensional real subspace of Schatten-3 can be linearly isometrically embedded into $L_3$. This resolves the case $p=3$ of Hanner’s inequality for Schatten classes, conjectured by Ball, Carlen and Lieb. We conjecture that similar isometric embedding is possible for any $p\ge 1$.