The anti-canonical cylinder criterion for weighted del Pezzo hypersurfaces

Let $S$ be a quasi-smooth, well-formed del Pezzo hypersurface of degree $d$ in the weighted projective space

$$S\subset\mathbb{P}(a_0,a_1,a_2,a_3),$$

where $S$ is defined over an algebraically closed field $\Bbbk$ of characteristic zero. An anti-canonical polar cylinder is a cylinder in $S$ whose boundary is supported by an effective anti-canonical divisor. The anti-canonical cylinder criterion. The surface $S$ admits an anti-canonical polar cylinder if and only if

$$d=a_i+a_j$$

for some $i\neq j$. This conjecture is motivated by known results for sporadic cases in the classification of quasi-smooth well-formed del Pezzo hypersurfaces in weighted projective spaces, but the asserted classification remains open in the stated generality.