Weak del Pezzo surfaces are characterized by the existence of $2$-tilting bundles

For $d\geq1$, a tilting bundle on a regular proper $d$-dimensional variety over a field is called $d$-tilting if its endomorphism algebra has global dimension $d$, the smallest possible value. We prove that, over any field, the existence of a $d$-tilting bundle implies that the anticanonical bundle