Weak geometric compatibility of T-junction extensions

Let $\mathscr{T} \in {\mathrm{W\mspace{-3mu}G\mspace{-2mu}A\mspace{-2mu}S}}$, and let $\mathbf{A}^{(1)},\mathbf{A}^{(2)} \in \mathscr{A}_{\mathbf{p}}$ satisfy

$$\operatorname{supp}_\Omega B_{\mathbf{A}^{(1)}}\cap \operatorname{supp}_\Omega B_{\mathbf{A}^{(2)}} \neq \nothing$$

and $\mathrm{v}_{j}(\mathbf{A}^{(1)}) \not\bowtie \mathrm{v}_{j}(\mathbf{A}^{(2)})$ for some $j\in\{1,\dots,d\}$. Let $\mathtt{T}\in\mathbb{T}_j$ be the T-junction inferred from the stated corollary, and put $k=\operatorname{pdir}(\mathtt{T})$.

Weak geometric T-junction conjecture. Then

$$\mathrm{v}_{k}(\mathtt{T}) \bowtie \mathrm{v}_{k}(\mathbf{A}^{(1)}) \quad\text{and}\quad \mathrm{v}_{k}(\mathtt{T}) \bowtie \mathrm{v}_{k}(\mathbf{A}^{(2)}).$$

The conjecture is presented as implying that weak geometric analysis-suitability is sufficient for weak dual compatibility. The supplied text does not establish whether this specific intermediate claim is proved, so its status remains open.