Exponential lower subdegree growth implies continuum many permutation-ends

Let $G$ be an infinite primitive permutation group whose subdegrees are all finite, and let $(m_r)$ be its lower subdegree sequence. Permutation-ends conjecture. If $(m_r)$ grows exponentially, then $G$ has $2^{\aleph_0}$ permutation-ends. This would, together with the preceding theorem, partition non-polynomial growth rates according to the number of permutation-ends; the claim is motivated by the fact that all known examples with exponential lower subdegree growth have infinitely many permutation-ends.