The interval decomposition conjecture for Ulam sequences U(1,n)

Let $U(a,b)$ denote the generalized Ulam sequence starting with integers $a,b$, where each subsequent term is the smallest integer representable as the sum of two distinct preceding terms in exactly one way. For $n\geq 4$, consider the sequence $U(1,n)$.

Interval decomposition conjecture. There exist integer coefficients $m_i,p_i,k_i,r_i$ such that, for every integer $n\geq 4$,

$$U(1,n)=\bigsqcup_{i=1}^{\infty}[m_i n+p_i,k_i n+r_i]\cap\mathbb{Z}.$$

The conjecture asserts a simple, uniform interval structure for all $U(1,n)$ with $n\geq4$, extending the explicitly observed initial patterns; it was confirmed computationally for thousands of terms but is not established in the supplied text.