Some chromatic height computations for homotopy fixed points of iterated topological Hochschild homology

For an $\mathbb{E}_{\infty}$-ring spectrum $R$ of chromatic height $k$, the trace method and the chromatic redshift imply that $T^n$-fixed point of $n$-fold iterated topological Hochschild homology of $R$ has chromatic height at most $k+n$. In this paper, we compute these heights for $H\mathbb{F}_p$, $H\mathbb{Z}$ and the $p$-complete connective image-of-$J$ spectrum $j_p$ for all primes and $n\geq1$, which are exactly $n-1$, $n$ and $n+1$, respectively. Our method is based on properties of Wilson spaces and relative topological Hochschild homology over $\mathrm{MU}$.