General irreducible-representation conjecture for proper biharmonic functions on SU(2)

For $n>1$, let $p,q\in{\mathbb C}^{n+1}$, and let $\pi_n$ be the $(n+1)$-dimensional irreducible representation of $\mathrm{SU}(2)$. For fixed indices $\alpha$ and $\beta$, define complex-valued functions on the unitary group by

$$P(z)=\sum_{j=1}^{n+1}p_j\pi^n_{j\alpha}(z),\qquad Q(z)=\sum_{k=1}^{n+1}q_k\pi^n_{k\beta}(z).$$

Let $W_Q=\\{z\in\mathrm{U}(2)\mid Q(z)\neq0\\}$ and define $f(z)=P(z)/Q(z)$ on $W_Q$. General irreducible-representation conjecture. If $\alpha\neq\beta$, then $f$ is proper biharmonic precisely when $p_nq_{n+1}-p_{n+1}q_n\neq0$ and

$$p_1q_{n+1}^n=q_n^{n-1}\bigl(n p_nq_{n+1}-(n-1)p_{n+1}q_n\bigr),\qquad q_1q_{n+1}^{n-1}=q_n^n,$$ $$p_2q_{n+1}^{n-1}=q_n^{n-2}\bigl((n-1)p_nq_{n+1}-(n-2)p_{n+1}q_n\bigr),\qquad q_2q_{n+1}^{n-2}=q_n^{n-1},$$

with the displayed pattern continuing through

$$p_{n-1}q_{n+1}^2=q_n\bigl(2p_nq_{n+1}-p_{n+1}q_n\bigr),\qquad p_{n-1}q_{n+1}=q_n^2.$$

The corresponding statement holds for the function induced on $\mathrm{SU}(2)$. This conjecture proposes a general pattern extending the explicitly investigated cases $n=2,3,4$; the source describes finding a general proof as a non-trivial exercise, and no resolution is supplied.