Joint ergodicity conjecture in the abelian setting

Let $D,\ell,L\in\mathbb{N}$, let $(X,\mathcal{X},\mu,U)$ be a $\mathbb{Z}^D$-system, and let $p_1,\ldots,p_\ell\in\mathbb{Z}^L[\mathbf{n}]$ be polynomials. The sequences are jointly ergodic when, for every $f_1,\ldots,f_\ell\in L^\infty(\mu)$, the corresponding multiple averages converge in $L^2(\mu)$ to the product of the integrals.

Joint ergodicity conjecture. The polynomials $p_1,\ldots,p_\ell$ are jointly ergodic for $(X,\mathcal{X},\mu,U)$ if and only if: (1) the product action $(U_{p_1(\mathbf{n})}\times\cdots\times U_{p_\ell(\mathbf{n})})_{\mathbf{n}\in\mathbb{Z}^L}$ is ergodic on $(X^\ell,\mathcal{X}^{\otimes\ell},\mu^{\otimes\ell})$; and (2) for every $1\leq i<j\leq\ell$, the difference action $(U_{p_i(\mathbf{n})}\circ U^{-1}_{p_j(\mathbf{n})})_{\mathbf{n}\in\mathbb{Z}^L}$ is ergodic on $(X,\mathcal{X},\mu)$.

The conjecture proposes necessary and sufficient conditions for joint ergodicity of polynomial actions in abelian systems, strengthening the paper's sufficient conditions in the 2-step nilpotent setting. The source attributes it to earlier formulations by Donoso, Frantzikinakis, Karageorgos, and Host, Donoso, Kra, and Shalom, and Krause.