Rigidity Conjecture for generalized Ulam sets

Let $U(a,N)$ be the generalized Ulam set with initial positive integers $a,N$. For a positive integer $L$, let $c$ range over the units $(\mathbb{Z}/L\mathbb{Z})^\times$. Let $A(x,y,s)$ denote the corresponding generalized interval component used in the decomposition, with integer parameters $m_{i_l},p_{i_l},k_{i_l},r_{i_l}$ and index parameter $s^{(l)}_{i_l}$.

Rigidity Conjecture. For every positive integer $a$, there exists an integer $L\geq1$ such that, for every $c\in(\mathbb{Z}/L\mathbb{Z})^\times$, there exists a positive integer $N_0\equiv c\pmod L$ such that for every $N\geq N_0$ satisfying $N\equiv c\pmod L$, one has a disjoint-union decomposition

$$U(a,N)=\bigcup_{l=1}^{\infty}\bigsqcup_{i_l=1}^{\infty}A\left(m_{i_l}N+p_{i_l},k_{i_l}N+r_{i_l},s^{(l)}_{i_l}\right),$$

with $m_{i_l}=k_{i_l}$ if and only if $l=1$. This conjecture proposes eventual, residue-class-dependent rigidity for generalized Ulam sets; the preceding discussion says that existing rigidity theorems only control initial segments, while numerical data suggests this stronger global structure.