Three-legs PT3 $qq$-character collision conjecture

Let $v_1$ and $v_2$ be spectral parameters, let $\mathsf{PT}_{\bar{4};\,\Bbox\,\Bbox\,\varnothing}(v_1)$ be the two-legs PT3 $qq$-character, let $\widetilde{\mathscr{Q}}_3(q_3v_2)$ be the screening charge, and let $\mathsf{f}(v_2/v_1)$ be the fusion factor. The three-legs PT3 $qq$-character is conjectured to arise from their collision:

Three-legs PT3 $qq$-character collision conjecture.

$$\mathsf{f}\left(\frac{v_{2}}{v_{1}}\right)\mathsf{PT}_{\bar{4};\,\Bbox\,\Bbox\,\varnothing}(v_{1})\widetilde{\mathscr{Q}}_{3}(q_{3}v_{2})\xrightarrow{v_{2}\rightarrow v_{1}} \mathsf{PT}_{\bar{4};\,\Bbox\,\Bbox\,\Bbox}(v_{1}).$$

This proposes that fusing a two-legs PT3 $qq$-character with the screening charge produces the three-legs character in the collision limit. The claim is presented as the main conjecture of the section; no proof or resolution is supplied.