The conjectural integral surgery triangle for rationally null-homologous knots

Let $\widehat{K}$ be a rationally null-homologous knot in $\widehat{Y}$, and let $\hat{\mu}$ and $\hat{\lambda}$ be the slope data used in the construction. Given $m\in\mathbb{Z}$ and sufficiently large $k$, define the sutured instanton homology objects appearing in the surgery triangle. The integral surgery triangle conjecture. There exists an exact triangle

\xymatrix{ \boldsymbol{\gamma}_{2\hat{\lambda}-(2m+2k-1)\hat{\mu}}\ar[rr]^{\pi_{m,k}(\hat{\mu})}&&\widehat{\mathbf{\Gamma}}_{m-1+2k}\ar[dl]\\ &\widehat{\mathbf{Y}}_{\hat{\lambda}-m\hat{\mu}}=I^{\sharp}(-\widehat{Y}_{-m}(\widehat{K}))\ar[ul]& }

Moreover, if $\hat{\lambda}-m\hat{\mu}$ is not the Seifert longitude $\lambda$, then

$$\pi_{m,k}(\hat{\mu})=\Psi_{+,m-1+2k}^{m+k}\circ\psi_{-,\mu}^{1}(\hat{\mu}^{\prime\prime})+\Psi_{-,m-1+2k}^{m+k}\circ\psi_{+,\mu}^{1}(\hat{\mu}^{\prime\prime}).$$

This is the proposed integral surgery formula in the rationally null-homologous setting; the source gives no evidence that it has been proved or refuted.