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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