Let $Y$ be oriented 3 manifold with torus boundary and let $\gamma_{j}$ (j=0,1,2) be three curves on its boundary with $\#(\gamma_{j}\cap \gamma_{j+1})=-1$. We denote by $Y_{j}$ the manifold obtained by gluing a solid torus to $Y$ with meridian $\gamma_{i}$. It was proved by Kronheimer-Mrowka-Ozvath-Szabo that there exists an exact triangle relating $\check{HM}(Y_{j};\mathbb{Z}_{2})$.
As mentioned in Kronheimer-Mrowka's book, one can expect there should be an exact triangle relating $\check{HM}(Y_{j};\mathbb{Z})$ and the proof should be very similar to the $\mathbb{Z}_2$ case. My question is that: is there any paper checking the detail of the proof for this $\mathbb{Z}$-coefficient exact triangle?
(Note that a $\mathbb{Z}$-coefficient exact triangle for Heegaard-Floer homology was proved by Ozvath-Szabo).
