Relative topological Hochschild homology conjecture for the Hopf-Galois extension X(n) to X(n+1)

Recall that the morphism of $\mathbb{E}_2$-ring spectra $X(n)\to X(n+1)$ is a Hopf-Galois extension with associated spectral Hopf algebra $\Omega S^{2n+1}$, regarded as the base space of the fibration

$$\Omega SU(n)\to \Omega SU(n+1)\to \Omega S^{2n+1}.$$

Relative THH conjecture. The relative topological Hochschild homology spectrum should satisfy

$$THH_{X(n)}(X(n+1))\simeq X(n+1)\wedge S^{2n+1}_+.$$

This is suggested by the preceding computation of $THH(X(n))$ and related results on Thom spectra; the source does not establish the relative equivalence.