The homological projective duality conjecture for the Cayley plane

Let $X$ be the Cayley plane, embedded in ${\mathbb P}^{26}$, and let its homological projective dual variety be the HPD variety associated with this embedding. The classical projective dual of $X$ is the Cartan cubic hypersurface in ${\mathbb P}^{26}$.

Homological projective duality conjecture. The homological projective dual variety of the Cayley plane is a non-commutative resolution of the Cartan cubic hypersurface in ${\mathbb P}^{26}$.

This claim interprets the vanishing of the HPD part for smooth hyperplane sections of the Cayley plane as evidence that the HPD variety is supported over the Cartan cubic. The statement is presented as a natural suggestion in the source, and no resolution is given there.