On the involutive Heegaard Floer homology of negative semi-definite plumbed 3-manifolds with $b_{1}=1$

In cite{MR1957829}, Ozsvath and Szabo use Heegaard Floer homology to define numerical invariants $d_{1/2}$ and $d_{-1/2}$ for 3-manifolds $Y$ with $H_{1}(Y;mathbb{Z})cong mathbb{Z}$. We define involutive Heegaard Floer theoret