Invariance of transfinite jumps under lower-level piecewise operators

Let $\beta$ and $\alpha$ be ordinals with $\beta < \alpha$, and let $\mathbf{X}$ be a represented space. Write $\mathbf{X}^{(\alpha)}$ for its $\alpha$-th jump and $\chi^{(\beta)}$ for the corresponding piecewise operator. Invariance conjecture. If $\beta < \alpha$, then

$$\chi^{(\beta)}\mathbf{X}^{(\alpha)} \cong \mathbf{X}^{(\alpha)}.$$

This asserts that applying a piecewise operator of strictly lower transfinite level does not change the represented space at the higher jump level. The supplied text gives no evidence about whether the claim has been proved or refuted.