Essential dimensions of polarized endomorphisms of certain algebraic surfaces

Let $f:X\to X$ be a polarized endomorphism of a smooth projective surface which is birationally ruled. We answer a question of Kollár and Zhuang, in the affirmative, on the incompressibility of $f$, under the assumption that $f$ is Galois and an explicit lower bound of deg$(f)$ depending only on $X$. We also give examples showing the optimality of such a lower bound.