Maslov grading compatibility conjecture for sutured Floer homology

Let (\Sigma,{\text{\boldmath \alpha}},{\text{\boldmath \beta}}) be a sutured Heegaard diagram for {\text{\boldmath \Sigma}}(D\times I,T). Write $\mathbb{T}_{\alpha}$ and $\mathbb{T}_{\beta}$ for the corresponding intersection tori, and let ${\bf M}_{SF}({\bf x},{\bf y})$ denote the relative Maslov grading whenever $\pi_2({\bf x},{\bf y})\neq\emptyset$. Maslov grading compatibility conjecture. There is a function

$${\bf M}:\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}\longrightarrow\mathbb{Q}$$

which satisfies

$${\bf M}({\bf x})-{\bf M}({\bf y})={\bf M}_{SF}({\bf x},{\bf y})$$

whenever ${\bf x},{\bf y}\in\mathbb{T}_{\alpha}\cap\mathbb{T}_{\beta}$ have $\pi_2({\bf x},{\bf y})\neq\emptyset$, and such that

$$E^{\infty}_{\delta=d}=SFH({\text{\boldmath $\Sigma$}}(D\times I,T))_{{\bf M}=d},$$

where the right-hand side is the subspace of SFH({\text{\boldmath \Sigma}}(D\times I,T)) consisting of homology classes representable as linear combinations of intersection points ${\bf x}$ with ${\bf M}({\bf x})=d$. This conjecture proposes that the spectral-sequence grading induced by the colored Jones complex agrees with the absolute Maslov grading on sutured Floer homology, extending the relative grading relation to the $E^{\infty}$ page.