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Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$ induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.

Given a null-homologous knot $K \subset M$ and a Seifert surface $F$ for $K$, we get a filtration (the Alexander grading) on these chain complexes. We can always find a basis for $\widehat{HF}(M)$ that is homogeneous in the Alexander grading.

Questions:

Is it known whether $\ker f$ has a basis that is homogeneous in the Alexander grading? Is it true that if $x,y \in \widehat{HF}(M)$ are Alexander-homogeneous classes of different Alexander grading such that $f(x),f(y) \neq 0$, then $f(x) \neq f(y)$?

Please provide a proof/reference or a counter-example.

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