Is the geography of Heegaard Floer homology restricted or the L-space conjecture false?

In a recent note F. Lin showed that if a rational homology sphere Y admits a taut foliation then the Heegaard Floer module HF⁻(Y) contains a copy of F[U]/U as a summand (arXiv:2309.01222). This implies that either the L-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. We verify that Lin's geography restriction holds for a wide class of rational homology spheres. Indeed, we show that the Heegaard Floer module HF⁻(Y) may satisfy a stronger geography restriction.