Noiselessness of the open core without definable integers

Let ${\mathscr{R}}$ be an expansion of $(\mathbb{R},<,+,\times)$, and let ${\mathscr{R}}^\circ$ be its open core. The structure ${\mathscr{R}}$ is type A when it has no dense definable subset of an interval admitting a definable ordering of order type $\omega$. Noiselessness conjecture. If ${\mathscr{R}}$ is type A, then ${\mathscr{R}}^\circ$ is noiseless. In particular, if ${\mathscr{R}}$ expands $(\mathbb{R},<,+,\times)$ and does not define $\mathbb{Z}$, then ${\mathscr{R}}^\circ$ is noiseless. This is presented as the main conjecture for expansions of the real ordered additive group; its relation to dense $\omega$-orders and definability of the integers motivates the claim.