Optimality of the fairy bread sandwich theorem

Let $k \le d$ be non-negative integers, and let $\pi$ be a permutation of $(d-1,\ldots,k-1)$. The paper considers flags of affine spaces and associated mass assignments, with the construction producing simultaneous bisections of the specified measures and functions.

Optimality conjecture. The result referred to as the fairy bread sandwich should be optimal for every such $k$, $d$, and permutation $\pi$.

The source provides no explanation of what “optimal” means beyond this statement and gives no evidence resolving it; the notation and the referenced result therefore require verification in the paper.