Eventual periodicity of combinatorial zero mutable Laurent polynomial counts

For positive integers $a,b$, let $\textup{ZMLP}_{\textup{comb}}(a,b)$ be the combinatorially defined set of pairs of partitions used to characterize zero mutable Laurent polynomials on rectangular triangles.

Eventual-periodicity conjecture.

1. For each $a\in\mathbb N$ there exists $b_0\in\mathbb N$ such that for all $b\geq b_0$, the number $|\textup{ZMLP}_{\textup{comb}}(a,b)|$ depends only on $b$ modulo $a$.
2. For each $k\in\mathbb N$ there exists $a_0\in\mathbb N$ such that for all $a\geq a_0$, $|\textup{ZMLP}_{\textup{comb}}(a,a+k)|$ depends only on $a$ modulo $k$.

The conjecture is motivated by computed tables for large rectangular triangles. It predicts eventual modular periodicity of the combinatorial upper bound, not directly of the number of zero mutable Laurent polynomials; the supplied text gives no proof.