The general structural conjecture for joinings of partially pro-nilsystem actions

Let $\boldsymbol{\Gamma}_i\leq\mathbb{Z}^D$ for $i=1,2,\ldots,r$ be subgroups with no pairwise inclusions, and let $n_1,n_2,\ldots,n_r\geq 0$ be integers. For each $i$, let $\boldsymbol{\mathbf{X}}_i$ be a $\boldsymbol{\Gamma}_i$-system, and write $\mathsf{Z}_{\mathrm{nil},n_i}^{\boldsymbol{\Gamma}_i}$ for the class of systems whose $\boldsymbol{\Gamma}_i$-subactions are direct integrals of $n_i$-step pro-nilsystems. A system is sated with respect to a family of classes when it has the corresponding satedness property for all possible joins of those classes.

General Structural Conjecture. Depending only on these data, there are finite families of pairs $(\Lambda_{i,j},m_{i,j})$ for $i=1,2,\ldots,r$, with $m_{i,j}\geq 0$ and $\Lambda_{i,j}\leq\mathbb{Z}^D$ properly containing $\Gamma_i$, such that if $\mathbf{X}_i\in\mathsf{Z}_{\mathrm{nil},n_i}^{\Gamma_i}$ and each $\mathbf{X}_i$ is sated with respect to all possible joins of classes $\mathsf{Z}_{\mathrm{nil},n}^{\Gamma}$, where $\Gamma\leq\mathbb{Z}^D$ and $n\geq 0$, then for any joining $\pi_i:\mathbf{Y}\longrightarrow\mathbf{X}_i$, the factors $\pi_i^{-1}(\Sigma_i)$ are relatively independent over

$$\pi_i^{-1}\Bigl(\bigvee_{j=1}^{k_i}\Phi_{i,j}\Bigr),$$

where $\Phi_{i,j}$ is the factor of $\mathbf{X}_i$ generated by the factor map to $(\mathsf{Z}_0^{\Gamma_i}\cap\mathsf{Z}_{\mathrm{nil},m_{i,j}}^{\Lambda_{i,j}})\mathbf{X}_i$.

This conjecture proposes a structural description of joinings when the acting subgroups are linearly dependent. In the special case $n_i=0$, it says that, under sufficient satedness, the joining factors are relatively independent over joins of factors arising from systems with trivial $\Gamma_i$-subaction and bounded-step pro-nilsystem structure. The result remains conjectural in the stated generality.