Non-interpretability of multiplication for Ulam expansions

Let $(a,b)$ be coprime, and let $\mathcal{U}_{a,b}$ denote the expansion of $(\mathbb{N},+,0,1)$ by the predicate for the associated Ulam sequence. Non-interpretability of multiplication. For every coprime $(a,b)$, the expansion $\mathcal{U}_{a,b}$ does not interpret $(\mathbb{N},+,\times)$. That is, there is no formula $\varphi(x,y,z)$ in the language $\{0,1,+,\mathrm{U}_{a,b}\}$ defining the graph of multiplication on an infinite domain. The conjecture is motivated by the purely additive character of the Ulam construction and the absence of an apparent mechanism for encoding multiplicative relations; its status is open.