Cubic generation conjecture for binary naive Bayes models

Let $G$ be a naive Bayes model with $r=2$ classes, and let $(p_{i_1 i_2 \cdots i_n})$ be its $n$-dimensional table. A flattening is any two-dimensional table obtained by partitioning the indices of this tensor into two groups; write $Q_G$ for the prime ideal of the model.

Cubic generation conjecture. The prime ideal $Q_G$ is generated by the $3\times 3$-subdeterminants of any two-dimensional table obtained by flattening the $n$-dimensional table $(p_{i_1 i_2 \cdots i_n})$.

The conjecture is motivated by computations showing cubic generators in the displayed examples. It asserts a uniform determinantal generating set for all binary naive Bayes models, but the supplied text gives no resolution or proof beyond those computations.