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

Let $m$ covariates index combinations of levels, and let $\Bj=(j_1,\dots,j_m)$ denote such a combination. Let $\bm{e}_{\Bj}$ be the array with $1$ at cell $(1,\Bj)$ and $-1$ at cell $(2,\Bj)$. Define ${\cal B}_{\Lambda(A_1 \otimes \dots \otimes A_m)}$ to consist of moves $z$ satisfying

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

A fiber has positive response marginals when the response-variable marginal for every combination of covariates is positive. Full-move connectivity conjecture. The set of moves ${\cal B}_{\Lambda(A_1 \otimes \dots \otimes A_m)}$ connects every fiber with positive response marginals for logistic regression with $m$ covariates. This is presented as the natural extension of the bivariate theorem. Parts of the bivariate proof generalize, but the arguments for Cases 3 and 5 do not currently extend to the multiple-covariate setting.