Polar de Rham theorem for smooth projective manifolds

Let $X$ be a smooth projective manifold of dimension $n$. The polar homology groups are denoted by $HP_k(X)$, the Dolbeault cohomology groups by $H^{n,n-k}_{\bar\partial}(X)$, and their dual cohomology groups by $HP^k(X)$ and $H^{0,k}_{\bar\partial}(X)$. The homomorphism induced by the pairing is

$$\rho:HP_k(X)\to H^{n,n-k}_{\bar\partial}(X).$$

Polar de Rham theorem. The mapping $\rho$ is an isomorphism. Equivalently,

$$HP^k(X)\cong H^{0,k}_{\bar\partial}(X).$$

The conjecture proposes that polar homology recovers Dolbeault cohomology for every smooth projective manifold. The surrounding discussion reports that examples support this assertion, but provides no resolution.