The oriented exact-triangle conjecture for framed real monopole Floer homology

Let $(L_+,L_-,L_0)$ be an oriented skein triple, and let $\mathbf{p}$ be a set of basepoints away from the resolutions. Denote by $\widetilde{\operatorname{HMR}}_*(L,\mathbf{p})$ the framed real monopole Floer homology of a based link. The oriented exact-triangle conjecture. The groups $\widetilde{\operatorname{HMR}}_*(L,\mathbf{p})$ fit into an oriented exact triangle

\begin{tikzcd} \widetilde{\operatorname{HMR}}_*(L_+,\mathbf{p}) && \widetilde{\operatorname{HMR}}_*(L_-,\mathbf{p}) \\\\ & \widetilde{\operatorname{HMR}}_*(L_0,\mathbf{p}) \arrow[from=1-1,to=1-3] \arrow[from=1-3,to=2-2] \arrow[from=2-2,to=1-1] \end{tikzcd}.

Here the basepoints are assumed to be away from the resolutions; the conjecture is the oriented analogue of the previously known unoriented exact triangle.