Restricted-move connectivity conjecture for positive-marginal fibers in multiple logistic regression

Let $m$ covariates index combinations of levels, and let ${\cal B}_{\Lambda(A_1 \otimes \dots \otimes A_m)}$ be the set of moves

$$z=\bm{e}_{\Bj_1}-\bm{e}_{\Bj_2}-\bm{e}_{\Bj_3}+\bm{e}_{\Bj_4},\qquad \Bj_1-\Bj_2=\Bj_3-\Bj_4.$$

Consider the subset for which every element of $\Bj_1-\Bj_2=\Bj_3-\Bj_4$ is in $\{-1,0,1\}$. A fiber has positive response marginals when the response-variable marginal for every combination of covariates is positive. Restricted-move connectivity conjecture. This subset of moves connects every fiber with positive response marginals for logistic regression with $m$ covariates. The source presents this as a stronger conjecture, even for $m=2$, motivated by examples suggesting that the move set can be further restricted; it remains open.