The explicit formula for the lowest spherical function in the second family

For each $n\geq 3$, let $\mu_n=\varpi_{n-1}$ and let $\widetilde\Psi^{\mu_n}_0(x)$ denote the lowest spherical function associated with the second family of $2\times2$ matrix-valued orthogonal polynomials for

$$(G,K)=\bigl(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2)\bigr).$$

Explicit lowest-spherical-function conjecture. For all $n\geq 3$,

$$\widetilde \Psi^{\mu_n}_0(x)=\begin{pmatrix} \dfrac{x+1}{2} & \dfrac{(n+1)x-2}{n-1} \\ \sqrt{x} & \dfrac{\sqrt{x}((n+3)x+n-5)}{2(n-1)} \end{pmatrix}.$$

This formula is an ansatz made from computations for small values of $n$, and the source gives no proof or resolution.