Web-basis conjecture for flamingo Specht modules

Let $r \geq 3$ and let $\mathscr{S} \subset \Pi(n,d,r)$ denote the collection of set partitions that can be made noncrossing by applying at most $r-3$ adjacent transpositions. For each set partition $\gamma \in \mathscr{S}$, order its blocks in any way to obtain a corresponding ordered set partition $\pi_\gamma$. Web-basis conjecture. The set

$$\{[\pi_\gamma]_r: \gamma \in \mathscr{S}\}$$

is linearly independent. The claim extends the preceding linear-independence theorem: for $r=3$, the relevant set consists of the noncrossing partitions and the result is already known, while the conjecture has been verified computationally for some higher values of $r$.