Copositivity conjecture for promotion maps

Let $\widetilde{\varphi}:\operatorname{Gr}(4,n)\to\operatorname{Gr}(4,n')$ be a promotion map. Promotion-map copositivity conjecture. The map $\widetilde{\varphi}$ is copositive, namely it restricts to a map

$$\widetilde{\varphi}:\operatorname{Gr}_{>0}(4,n)\to\operatorname{Gr}_{>0}(4,n').$$

This conjecture is attributed in the source to the cited work denoted VRC and is intended to explain positivity of substitution maps; its resolution is not stated.