Filling-count conjecture for twist-spun Legendrians

Let $\rho$ be the twisting monodromy, let $\rho^k$ denote its $k$-fold iterate, and let $\Sigma_{\rho^k}(\lambda(2,n))$ be the corresponding twist-spun Legendrian. Let $f(k)$ be the function defined earlier in the source. Filling-count conjecture. There are exactly $f(k)$ exact Lagrangian fillings of $\Sigma_{\rho^k}(\lambda(2,n))$. The conjecture is part of the paper's proposed relationship between twist-spun Legendrians, sheaf moduli, and folded cluster structures; no resolution is supplied.