Paper Boat zone partition conjecture

Let $\Phi$ be an affine root system with coweight lattice $\Lambda^\vee$, dominant coweights $(\Lambda^\vee)^+$, and finite Weyl-group alcove set $\mathcal{F}$. For $a,b\in\mathcal{F}$, $\sigma\in\Omega$, and dominant coweights $\lambda,\mu$, let $PB(a,\lambda,b,\sigma)$ be the corresponding Paper Boat and let $W_\mathrm{f}$ be the finite Weyl group.

Paper Boat zone conjecture. For every affine Weyl group there is a partition $\mathcal{Z}$ of $(\Lambda^\vee)^+$ with at most $3^n$ parts such that, for every $Z\in\mathcal{Z}$ and all $\lambda,\mu\in Z$,

$$|PB(a,\lambda,b,\sigma)|=|PB(a,\mu,b,\sigma)|.$$

Thus only finitely many Paper Boat sizes would need to be computed for fixed affine type, uniformly over the parameters $a,b,\sigma$.