The $L^2$-Stokes conjecture for complex projective varieties

Let $V$ be a complex projective variety, let $M=V\setminus\operatorname{sing} V$ be its regular part, and equip $M$ with the metric induced by a smooth Kähler metric on projective space. Let $d_{\min}$ and $d_{\max}$ denote respectively the minimal and maximal closed extensions of the de Rham differential on square-integrable forms. The $L^2$-Stokes conjecture. The minimal and maximal extensions coincide, $d_{\min}=d_{\max}$. This would give uniqueness of the closed extension of the de Rham differential and extend the $L^2$-Stokes theorem from smooth manifolds to singular complex projective varieties. The statement is explicitly identified in the source as the main open problem; it is known for complex surfaces, but remains open in general.