Necessary sampling-condition conjecture for tensor dynamical sampling

Let $\mathcal{F} \in \mathbb{C}^{m \times p \times n}$ be a signal, let $\Psi$ denote its samples under the stated tensor-product dynamical evolution, and let $\Omega \subseteq [m] \times [p] \times [n]$ be the sampling set. Necessary condition conjecture. A necessary condition for the successful recovery of $\mathcal{F}$ from $\Psi$ is

$$\bigcup_{(i,j,k)\in\Omega} \{j\} = [p].$$

The preceding theorem establishes this condition for lattice sampling sets of the form $\Omega=I\times J\times[n]$, where $I\subseteq[m]$ and $J\subseteq[p]$: recovery is impossible when $J\neq[p]$. The conjecture extends that necessary condition to general sampling sets and is proposed for future work.