Solenoidal Hodge conjecture

Let $M$ be a compact Kähler manifold of complex dimension $n$. A complex immersed solenoid in $M$ is a solenoid whose leaves are mapped to complex immersed submanifolds; such a solenoid of dimension $k=2(n-p)$ represents a class in $H_{n-p,n-p}(M)=H^{p,p}(M)^*\subset H_k(M,\mathbb{R})$. Solenoidal Hodge conjecture. Every class in $H^{p,p}(M)$ is represented by a complex immersed solenoid of dimension $k=2(n-p)$. This is the proposed real-homology analogue of the Hodge conjecture, replacing integral classes by real classes and algebraic cycles by complex immersed solenoids.