Numerical rigidity conjecture for the intervals of U(1,n)

For each $n$, write the generalized Ulam set $U(1,n)$ as its unique disjoint union of interval components

$$U(1,n)=\bigsqcup_{i=1}^{\infty}A_i(n),\qquad A_i(n)=[a_i(n),b_i(n)]\cap\mathbb{Z}.$$

Numerical rigidity conjecture. There exists an integer $N_0>1$ such that, for every $n\geq N_0$, the endpoints are given by

$$a_i(n)=(n+B)m_i+\varepsilon_i,$$ $$b_i(n)=(n+B)p_i+\delta_i,$$

where $m_i,p_i,\varepsilon_i,\delta_i$ do not depend on $n$, $B,\varepsilon,\delta>0$ are real constants, $m_i,p_i$ are integers, and $|\varepsilon_i|<\epsilon$, $|\delta_i|<\delta$. This predicts eventual linear dependence, with uniformly bounded endpoint errors, for the interval decomposition of $U(1,n)$; it is presented as based on numerical data and is not proved in the supplied text.