Reichardt diagonalizing sequence convergence conjecture

Let $n\in\mathbb{N}$, set $p=2n+1$ and $\theta_p=\frac{\pi}{p}$, and let $D_j(\theta_p)=\operatorname{diag}(\lambda_{j0},\lambda_{j1})$ for $j=1,\ldots,n$, where

$$\lambda_{j0}=\omega^{(-1)^j j},\qquad \lambda_{j1}=(-1)^{j+1}\omega^{(-1)^{j+1}j},\qquad \omega=e^{i\theta_p/2}.$$

For $U_0\in\operatorname{U}(2)$, define the diagonalizing sequences recursively by

$$U_{k+1}^{(n)}=A_p(U_k;\theta_p)=Q_nU_k^{(-1)^n}P_n,$$

where $P_0=Q_0=I$ and

$$P_{j+1}=D_{j+1}(\theta_p)U_k^{(-1)^j}P_j,\qquad Q_{j+1}=Q_jU_k^{(-1)^j}D_{j+1}(\theta_p).$$

Diagonalizing sequence conjecture. For every $U_0\in\operatorname{U}(2)$, the sequences satisfy

$$\left|(U_{k+1}^{(n)})_{21}\right|=\left|b_{k+1}^{(n)}\right|=\left|b_k^{(n)}\right|^{2n+1}=\left|b_k^{(n)}\right|^p.$$

This exact-power relation would establish the claimed rapid diagonalization of arbitrary single-qubit unitaries by the recursively constructed sequences. The supplied text does not state whether the conjecture has been proved or disproved.