Lefschetz-type preservation for scattering diagrams

Let $\f{D}_{\operatorname{in}}$ be an initial scattering diagram of the form specified in the source, with incoming wall functions determined by Laurent polynomials $p_i(t)$. A scattering diagram $\f{D}$ represents $\operatorname{Scat}(\f{D}_{\operatorname{in}})$ when it is a consistent completion of the initial diagram. Scattering-diagram Lefschetz conjecture. If each $p_i(t)$ is of Lefschetz type, then one can choose $\f{D}$ representing $\operatorname{Scat}(\f{D}_{\operatorname{in}})$ so that all walls have the prescribed form and every associated polynomial $p_{\f{d}}(t)$ is of Lefschetz type. The corresponding positivity, bar-invariance, and pre-Lefschetz properties are proved in the paper, while this stronger Lefschetz-type preservation statement is used as a conjectural input for results about quantum theta coefficients.