The exponential multiplicity conjecture for self-overlapping polygons

Let a self-overlapping polygon have boundary length $2n$, and let its multiplicity be the number of distinct distorted disks it bounds. Exponential multiplicity conjecture. There exist strictly positive constants $A$ and $B$ such that every such polygon has multiplicity at most

$$AB^{2n}.$$

The bound would imply $w_{2n}\leq AB^{2n}\hat w_{2n}\leq4A(3B)^{2n}$ and hence $\ln w_{2n}=O(n)$; establishing this linear exponential-growth bound is left open.