A chromatic vanishing result for TR

In this note, we establish a vanishing result for telescopically localized $\mathrm{T}\mathrm{R}$. More precisely, we prove that $T(k)$-local $\mathrm{T}\mathrm{R}$ vanishes on connective $L_n^{p,f}$-acyclic ${\mathbf{E}}_1$-rings for every $1\le k\le n$ and deduce consequences for connective Morava K-theory and the Thom spectra $y(n)$. The proof relies on the relationship between $\mathrm{T}\mathrm{R}$ and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive $\mathrm{\infty }$-categories which was recently established by Córdova Fedeli.