Gap-distribution conjecture for Ulam words

Let $\pi:S[\{0,1\}]\to\mathbb{Z}_{\geq0}$ interpret a binary word as an integer, and let $u_1<u_2<\ldots<u_{k_n}$ be the ordered elements of $\pi(\mathscr{U}_n)$. Define

$$p_n:\mathbb{Z}_{\geq1}\to[0,\infty),\qquad g\mapsto\frac{\#\{i\mid u_{i+1}-u_i=g\}}{k_n-1}.$$

Let $\mu_g(n)$ be the mean of the probability measure $p_n$. Gap-distribution conjecture. As $n\to\infty$, the functions $p_n$ converge pointwise to a probability measure $p:\mathbb{Z}_{\geq1}\to[0,\infty)$, and

$$\mu_g(n)=\Theta(n^{3/10});$$

indeed, there may be a constant $c\approx1.9$ such that

$$\mu_g(n)=cn^{3/10}+o(1).$$

The conjecture is supported by the paper's numerical data and, if true, implies the conjectured asymptotic density through a proved comparison between the reciprocal density and the mean gap. Its general resolution remains open.