The surface-construction conjecture for filtered deformations of elliptic algebras

Let $B_{V,\Psi}$ be an elliptic algebra, and let a filtered deformation mean a nontrivial filtered deformation of $B_{V,\Psi}$. Let $A^+_M$ denote the Rees algebra associated with a sheaf $M$ on a noncommutative del Pezzo surface. A sheaf $M$ is rigid, unobstructed, and torsion-free in the usual deformation-theoretic sense. Surface-construction conjecture. Any nontrivial filtered deformation of any elliptic algebra $B_{V,\Psi}$ has Rees algebra of the form $A^+_M$ for some rigid, unobstructed, torsion-free sheaf $M$ on a noncommutative del Pezzo surface. The conjecture is supported by computations in several line-bundle, slope-zero, and nonintegral-slope cases, but the general twisted case remains out of reach.