The N-filling formula for unoriented knot Floer homology

Let $Y$ be a $3$-manifold and $K\subset Y$ a knot, with meridian $\mu$. Let $N$ be the twisted $I$-bundle over the Klein bottle, and let $Y_N^\mu$ be the N-filling of $Y\setminus\operatorname{int}N(K)$ with respect to $\mu$, obtained by gluing $N$ so that its homological longitude maps to $\mu$. Let $HFK'_2(Y,K)$ be the proposed unoriented knot Floer homology, and let $\widehat{HF}$ denote Heegaard Floer homology. N-filling conjecture. There is an isomorphism

$$\widehat{HF}(Y_N^\mu)\cong HFK'_2(Y,K)\oplus\widehat{HF}(Y)\oplus\widehat{HF}(Y).$$

This conjecture connects the proposed knot Floer invariant with the $N$-filling construction used in the study of graph-manifold L-space questions; the supplied source gives no resolution status.