The asymptotic period conjecture for coefficients of U(1,X)

Let $U(1,X)$ denote the polynomial-ring analogue of the Ulam sequence, and let $a_i,b_i,c_i,d_i$ be its coefficients. For real $X$, reduction modulo $3X+\lambda'$ is taken using representatives in an interval of length $3X+\lambda'$. Asymptotic period conjecture. There exist real numbers $\lambda'\approx0.417031$, $\sigma_1\approx1.86$, and $\sigma_2\approx-1.3$ such that, for every $\epsilon>0$, if $i$ is sufficiently large, then

$$a_iX+b_i,\ c_iX+d_i\mod(3X+\lambda')\in\left(X+\frac{\lambda'}{3}-\epsilon,X+\sigma_1+\epsilon\right)\cup\left(2X+\sigma_2-\epsilon,2X+\frac{2\lambda'}{3}+\epsilon\right).$$

This conjecture gives a more precise form of the observed period phenomenon for the block endpoints of $U(1,n)$; the source reports numerical evidence but no proof.