The maximal-vertex realization conjecture for Newton–Okounkov bodies on surfaces

Let $S$ be a smooth projective algebraic surface, let $D$ be a big divisor on $S$, and write $\rho=\rho(S)$. A smooth projective surface $\tilde S$ with a proper projective morphism $\tilde S\to S$ and a flag Y_\bullet=\\{\tilde S\supset E\supset\{p\}\} determine the Newton–Okounkov body $\Delta_{Y_\bullet}(D)$. Maximal-vertex realization conjecture. There is such a surface $\tilde S$, morphism, and flag for which $\Delta_{Y_\bullet}(D)$ has exactly $2\rho+2$ vertices. The upper bound of $2\rho(S)+2$ is proved in the paper, while realization of the bound in this generality is the conjectural part.