Cheltsov–Park–Won conjecture on polarized cylinders in log del Pezzo surfaces

Let $S$ be a log del Pezzo surface, meaning a del Pezzo surface with at worst klt singularities. Let

$$\mathrm{Amp}^{\mathrm{cyl}}(S):=\{H\in\mathrm{Amp}(S)\mid S\text{ contains an }H\text{-polar cylinder}\}$$

be its cylindrical ample set, where $\mathrm{Amp}(S)$ is the ample cone. Cheltsov–Park–Won conjecture.

$$-K_S\in\mathrm{Amp}^{\mathrm{cyl}}(S)\quad\Longleftrightarrow\quad \mathrm{Amp}^{\mathrm{cyl}}(S)=\mathrm{Amp}(S).$$

This conjecture asks whether the existence of an anticanonically polarized cylinder on a log del Pezzo surface is equivalent to every ample polarization admitting a polarized cylinder. Its status is not resolved in the supplied source context.