The Lin geography restriction conjecture for Heegaard Floer homology

Let $Y$ be a rational homology sphere. Say that $\operatorname{HF}^-(Y)$ satisfies the Lin geography restriction when it has either no $U$-torsion or an $\mathbb{F}[U]/U\simeq\mathbb{F}$ summand.

Lin geography restriction conjecture. The Heegaard Floer module $\operatorname{HF}^-(Y)$ satisfies the Lin geography restriction for all rational homology spheres $Y$.

Lin's theorem shows that a rational homology sphere admitting a taut foliation has an $\mathbb{F}[U]/U$ summand in its Heegaard Floer module. The conjecture extends the resulting restriction to all rational homology spheres and is supported by surgery-formula calculations for a broad class of examples.