The two-dimensional Schatten-to-L_p isometric embedding conjecture

For $p\geq 1$, let $\mathcal{S}_p$ denote the Schatten-$p$ class, and let $A,B\in\mathcal{S}_p$. The real space $\operatorname{span}_{\mathbb{R}}\{A,B\}$ is equipped with the Schatten-$p$ norm.

Two-dimensional Schatten embedding conjecture. For any $p\geq 1$ and $A,B\in\mathcal{S}_p$, the real space

$$\operatorname{span}_{\mathbb{R}}\{A,B\}$$

can be linearly isometrically embedded into $L_p$.

If true, this would imply the noncommutative Hanner inequalities throughout the ranges $p\geq 2$ and $1\leq p\leq 2$, and would reduce inequalities depending only on norms of elements in a plane to the commuting or diagonal case. The conjecture is known for $p=3$ by the main result of the paper, while the general case remains open.