The diagonal extremality conjecture for central sections of the cube

Let $Q_n=[-1,1]^n$ be the $n$-dimensional cube, and call a central section locally extremal if its volume is a local maximum or minimum of the central section-volume function. A central section is diagonal when its normal vector is proportional to a vector with all coordinates equal in absolute value.

Diagonal extremality conjecture. All locally extremal central sections of $Q_n$ are diagonal for each $n\geq 2$.

This conjecture concerns the unresolved possibility of locally extremal non-diagonal critical sections. The paper constructs non-diagonal critical central sections with no zero coordinates in every dimension $n\geq4$, but shows that these examples are saddle points rather than locally extremal sections.