The Section Property for Calabi--Yau and Fano Fermat hypersurfaces

Let $X$ be a Fermat hypersurface of degree $d$ and dimension parameter $r$, in the notation of the paper. Let $(\star)$ denote the distinguished-marking condition defined there. Fermat hypersurface conjecture. If

$$d\leq r+2,$$

so that the Fermat hypersurface is Calabi--Yau or Fano, then $X$ satisfies Condition $(\star)$. The paper states that it cannot determine whether other Fermat hypersurfaces satisfy this condition; the conjecture therefore isolates the Calabi--Yau and Fano range.