The Hodge conjecture in Bott-Chern cohomology

Let $Y$ be a compact complex manifold. Define

$$H^{p,p}_{BC}(Y;\Q):=PD^{-1}(H^{BC}_{2(n-p)}(Y;\Q))\cap H^{p,p}_{BC}(Y).$$

A formal linear combination of irreducible holomorphic subvarieties of dimension $p$ with rational coefficients is a holomorphic $p$-chain with $\Q$-coefficients.

The Hodge conjecture in Bott-Chern cohomology. Every element of $H^{p,p}_{BC}(Y;\Q)$ is representable by holomorphic $p$-chains with $\Q$-coefficients.

This asks for an analogue of the classical Hodge conjecture for Bott-Chern cohomology on compact complex manifolds. The statement extends the algebraic-cycle formulation to holomorphic chains and is open in the generality stated.