The quasiconcavity conjecture for Burkholder functions

For $p\geqslant n/2$ and $\lambda\geqslant\left|1-n/p\right|$, define the Burkholder matrix function

$$\mathbf E(A)=\left[\pm\det A-\lambda|A|^n\right]|A|^{p-n},\qquad A\in\mathbb R^{n\times n}.$$

The source also defines the planar Burkholder function $\mathbf B_p(\xi,\zeta)$ and calls these functions Burkholder functions. Quasiconcavity conjecture. Burkholder functions are quasiconcave.

The preceding theorem establishes rank-one concavity, but rank-one concavity does not generally imply quasiconcavity. The conjecture asks whether this implication holds for this distinguished family of integrands.