  Tilting bundles $\mathcal{T}$ on a weighted projective line $\mathbb{X}$ have
been intensively studied by representation theorists since they give rise to a
derived equivalence between $\mathbb{X}$ and the finite dimensional algebra End
$\mathcal{T}$. A classical result states that if End $\mathcal{T}$ is
hereditary, then $\mathbb{X}$ is Fano and conversely, for every Fano weighted
projective line, there exists a tilting bundle $\mathcal{T}$ with End
$\mathcal{T}$ hereditary. In this paper, we examine the question of when a
weighted projective surface has a tilting bundle whose endomorphism ring is
2-hereditary in the sense of Herschend-Iyama-Oppermann. It is natural to
conjecture that they are the almost Fano weighted surfaces, weighted only on
rational curves, and we give evidence to support this.
