Asymptotic density conjecture for Ulam words

Let $\mathscr{U}_n$ denote the set of Ulam words of length $n$, and define

$$\rho(n):=\frac{\#\mathscr{U}_n}{2^n}.$$

Ulam-word density conjecture. The density satisfies

$$\rho(n)=\Theta(n^{-3/10}).$$

This conjecture is based on the enlarged computational data set in the paper and is presented as a proposed alternative to the earlier conjecture that the density tends to a positive constant. No proof or resolution is given.