Gibbs's concentration conjecture for the Ulam sequence U(1,2)

Let $U(1,2)$ be the Ulam sequence, and for a positive real number $\lambda$ interpret reduction modulo $\lambda$ by taking representatives in an interval of length $\lambda$. Gibbs's concentration conjecture. There exists a real number $\lambda_2\approx2.443442967784743433$ such that for all $\epsilon>0$, for $K$ sufficiently large,

$$U(1,2)\cap[K,\infty)\mod\lambda_2\subset\left(\frac{\lambda_2}{3}-\epsilon,\frac{2\lambda_2}{3}+\epsilon\right).$$

The conjecture concerns the apparent concentration of the residues of $U(1,2)$ in the middle third modulo a distinguished real period. It was confirmed computationally for the first trillion terms, but the asymptotic assertion remains open.