Tameness under bounded gaps for Ulam expansions

Let $(a,b)$ be coprime, let $U(a,b)$ be the associated Ulam sequence, and write its consecutive gaps as $g_k$. Condition (R3) means that there exists $B$ such that all gaps $g_k$ are at most $B$. Tameness under bounded gaps. If $U(a,b)$ satisfies (R3), then:

1. The expansion $\mathcal{U}_{a,b}$ is NIP.
2. For any formula $\varphi(x;y)$ in the language $\{0,1,+,\mathrm{U}_{a,b}\}$ with $|x|=1$, the VC-dimension of the family $\{\varphi(\mathbb{N};b):b\in\mathbb{N}^{|y|}\}$ is finite.

Bounded gaps are expected to force model-theoretic tameness, but the source states that complete proofs are not available under (R3) alone. The conjecture is open.