The anticanonical cylinder conjecture for log del Pezzo surfaces

Let $S$ be a log del Pezzo surface. Let $\mathrm{Amp}(S)$ be its ample cone in $\operatorname{Pic}(S)\otimes\mathbb{R}$, and let $\mathrm{Amp}^{cyl}(S)$ be the set of ample classes $H$ for which $S$ has an $H$-polar cylinder. A cylinder is $(-K_S)$-polar when it is polarized by the anticanonical class $-K_S$. Anticanonical cylinder conjecture. The surface $S$ has a $(-K_S)$-polar cylinder if and only if $\mathrm{Amp}^{cyl}(S)=\mathrm{Amp}(S)$. This proposes that the existence of an anticanonically polarized cylinder is equivalent to every ample polarization admitting a cylinder; the cited preceding results provide evidence for this supposition, but its general validity is not established here.