Conjecture that subexponential exponential growth is equivalent to monadic NIP

Let $M$ be an $\omega$-categorical structure, and let $f_M(n)$ denote its growth rate. A structure is monadically NIP when all expansions of it by finitely many unary predicates are NIP.

Monadic NIP growth conjecture.

$$f_M(n)<c^n$$

for some $c\in\mathbb{R}$ if and only if $M$ is monadically NIP.

The source notes that non-monadically NIP structures have growth faster than exponential, giving partial progress toward the converse; the equivalence itself remains open there.