W1 vanishing on the Hartogs triangle in C2

Determine whether the Sobolev cohomology group H^{0,1}_{W^1}(T) vanishes for the Hartogs triangle T = {(z, w) ∈ C^2 : |z| < |w| < 1}.

Background

The Hartogs triangle is a classical non-Lipschitz pseudoconvex domain where many regularity phenomena are subtle. While recent work proves vanishing of H^{0,1}_{W^{k,p}}(T) for k ∈ N and p > 4, the endpoint L^2 case (W^{1,2} = W^1) is unresolved.

The authors state the problem and explicitly note that the L^2 Sobolev case remains unknown; they also contrast this with related non-vanishing results on Hartogs triangles in CP^2.