Ulam dichotomy for expansions of Presburger arithmetic

Let $(a,b)$ be coprime, and let $\mathcal{U}_{a,b}$ be the expansion of $(\mathbb{N},+,0,1)$ by the predicate for the associated Ulam sequence. Ulam dichotomy. For every coprime $(a,b)$, exactly one of the following holds:

1. The expansion $\mathcal{U}_{a,b}$ is a reduct of a definitional expansion of Presburger arithmetic, possibly with countably many parameters, and in particular is NIP, dp-minimal, and does not interpret $(\mathbb{N},+,\times)$.
2. The expansion $\mathcal{U}_{a,b}$ interprets $(\mathbb{N},+,\times)$, and in particular has the independence property and is model-theoretically wild.

Moreover, case 2 does not occur: every Ulam expansion falls into case 1. This conjecture proposes a dichotomy between Presburger-like tameness and interpretation of full arithmetic, while asserting that only the tame case occurs; its status is open.