Flamingo-web injection conjecture for three-row increasing tableaux

A normal plabic graph is a plabic graph whose boundary vertices are black of degree $1$, whose internal black vertices have degree $3$, and in which no two vertices of the same color are joined by an edge. A flamingo web is a normal plabic graph in which every interior face has at least $6$ vertices and every white vertex has degree at least $3$. Let $\mathrm{FW}(n,k)$ be the set of flamingo webs with $n$ boundary vertices for which the number of white vertices minus the number of black interior vertices is $k$. Let $\mathsf{trip}_{i,3}$ denote the corresponding trip digraphs, let $\mathsf{prom}_i$ denote $K$-promotion on the associated increasing tableaux, and let $\tau(W)$ denote the tableau associated to a web $W$. Flamingo-web injection conjecture. Each $W\in\mathrm{FW}(q,k)$ has a distinct pair $\mathsf{trip}_{1,3},\mathsf{trip}_{2,3}$ of trip digraphs. Moreover, for each $W$, there is a unique tableau $\tau(W)\in\operatorname{Inc}^q(3\times(q-2k))$ such that

$$\mathsf{trip}_{i,3}(W)=\mathsf{prom}_i(\tau(W))\qquad\text{for }i=1,2.$$

This yields an injection $\tau:\mathrm{FW}(q,k)\to\operatorname{Inc}^q(3\times(q-2k))$ intertwining web rotation with $K$-promotion. The conjecture is motivated by a conjectural connection between flamingo webs and Specht modules, while the stated relationship with three-row tableaux and $K$-promotion remains open.