Structure decomposition conjecture for the end stabilisers

Let $B_{i,\infty}^\pm$ be the stabiliser of the corresponding end, with subgroups $B_W^\pm$, $U_i$, and $W^{even}$ as defined in the paper. Structure decomposition conjecture. For $i=1,2$, one has

$$B_{i,\infty}^\pm=B_W^\pm U_iW^{even}=B_W^\pm W^{even}U_i=U_iW^{even}B_W^\pm=U_iB_W^\pm W^{even}=W^{even}U_iB_W^\pm=W^{even}B_W^\pm U_i.$$

The paper states this as a consequence that would follow from the preceding transitivity conjecture and supplies a conditional proof using subgroup-normalisation lemmas. Since the preceding conjecture is not established, this decomposition remains open here.