Happel–Unger conjecture on connectedness of tilting graphs

Let $\mathcal{H}$ be a connected hereditary abelian category. Let $\mathbb{X}$ be a weighted projective line and $H$ a finite-dimensional hereditary algebra. Assume that $\mathcal{H}$ is derived equivalent to $\operatorname{coh}\mathbb{X}$ with $\mathbb{X}$ tubular or wild, or to $\operatorname{mod} H$ with $H$ wild. Let $\mathcal{G}_t(\mathcal{H})$ denote the tilting graph of $\mathcal{H}$, whose vertices are the isomorphism classes of basic tilting objects and whose edges join objects differing by precisely one indecomposable direct summand.

Happel–Unger conjecture. The graph $\mathcal{G}_t(\mathcal{H})$ is connected if and only if $\mathcal{H}$ contains no nonzero projective objects.

The conjecture gives a criterion for connectedness in the tubular and wild cases of Happel’s classification of hereditary abelian categories with tilting objects. The supplied text does not state a resolution.