Oppositely transformed matrices can be made hollow and almost hollow

Let $A,B\in\mathbb{R}^{n\times n}$ satisfy

$$\operatorname{trace} A=\operatorname{trace} B=0.$$

A matrix is hollow if all its diagonal entries are zero, and almost hollow if it has the corresponding property defined earlier in the paper. The transformations considered below act oppositely on the two matrices.

Hollowisation conjecture. There exists an orthogonal matrix $V\in\mathbb{R}^{n\times n}$ such that $V^TAV$ is hollow and $VBV^T$ is almost hollow.

Here the transformation applied to $A$ is the inverse, equivalently the adjoint, of the transformation applied to $B$, unlike in the preceding simultaneous-transformation proposition. The paper states this variant because it appears to hold numerically, but does not provide a proof.