Palka's height bound conjecture for singular del Pezzo surfaces

Let $\overline{X}$ be a singular del Pezzo surface. The height bound conjecture. If $\operatorname{char}\mathscr{k}\neq 2,3$, then

$$\operatorname{ht}(\overline{X})\leqslant 4.$$

The height is a bounded invariant intended to organize the classification of del Pezzo surfaces of rank one through the $\mathbb{P}^1$-fibrations realizing it. The bound is stated as a result to be proved in a forthcoming article, so its resolution is not established in the supplied text.