L-spaces and F-summands

Problem: Prove that if $S^3_r(K)$ is not an L-space then its Heegaard Floer homology has a direct $\mathbb{F}$-summand.

The L-space conjecture predicts that if an irreducible closed 3-manifold $Y$ is not an L-space then $Y$ admits a coorientable taut foliation. Recently, Lin proved in a short note that if $Y$ admits a coorientable taut foliation then its Heegaard Floer homology $\mathit{HF}^+(Y)$ has a direct $\mathbb{F}$-summand. 

For background: recall that $\mathit{HF}^+(Y)$ is a module over $\mathbb{F}[U]$ where $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$. A direct $\mathbb{F}$-summand of $\mathit{HF}^+(Y)$ refers to a direct summand isomorphic as an $\mathbb{F}[U]$-module to $\mathbb{F}[U]/U$.

Lin's result therefore gives a potential way to disprove the L-space conjecture: find an irreducible non-L-space whose Heegaard Floer homology does not have a direct $\mathbb{F}$-summand. Alternatively, one could try to prove that Lin's result cannot possibly give a counterexample by showing that every non-L-space has a direct $\mathbb{F}$-summand in its Floer homology. The problem asks to do precisely this for 3-manifolds $S^3_r(K)$ arising via surgery on a knot $K\subset S^3$. In particular, this means showing that $\mathit{HF}^+(S^3_r(K))$ has a direct $\mathbb{F}$-summand for every rational $$r<2g(K)-1$$ when $K$ is an L-space knot, and for every rational $r$ when $K$ isn't an L-space knot. This is a tractable problem that should be approachable using the surgery formula in Heegaard Floer homology or Hanselman's minus version of the immersed curves framework for surgeries.

Update: Antonio Alfieri and Fraser Binns have proved the result stated in the problem here.

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