Cheltsov's cylindrical ample cone conjecture for log del Pezzo surfaces

Let $S$ be a log del Pezzo surface. The cylindrical ample cone conjecture. If the cylindrical ample cone

$$\operatorname{Amp^{\mathrm{cyl}}}(S)$$

contains the anticanonical class, then

$$\operatorname{Amp^{\mathrm{cyl}}}(S)=\operatorname{Amp}(S).$$

Equivalently, the cylindrical ample cone contains the anticanonical class if and only if it equals the ample cone. The conjecture concerns when the existence of an anticanonical polar cylinder forces every ample polarization on a log del Pezzo surface to admit a polar cylinder; the source does not provide resolution evidence for the general statement.