Complex $\mathrm{BC}_n$ constant term conjecture

Let $n\equiv\zeta\pmod 4$ with $\zeta\in\{0,1\}$, let $u\in\mathbb C$ satisfy

$$\min\{\operatorname{Re}(1+2b+(n-1)u),\operatorname{Re}(1+nu/2)\}>0,$$

and let $\tau_{ij}$ and $\sigma_{ij}$ be signatures satisfying the paper's condition. Write $x=\cos(\pi u/2)$. Complex $\mathrm{BC}_n$ constant term conjecture. There exists a polynomial $P_n(x)$, independent of $a$ and $b$, with $P_n(1)=1$, such that

$$\begin{aligned} \operatorname{CT}&\bigg[\prod_{i=1}^n (1-x_i^{\pm})^a(1-x_i^{\pm2})^b\prod_{1\leq i<j\leq n}\Big(1-(x_ix_j)^{\sigma_{ij}}\Big)^u\Big(1-\Big(\frac{x_i}{x_j}\Big)^{\tau_{ij}}\Big)^u\bigg]\\ &=x^{n-\zeta}P_n(x^2)\frac{\Gamma(1+nu/2)}{\Gamma(1+(n-1)u/2)\Gamma^n(1+u/2)}\prod_{i=1}^{n-1}\frac{\Gamma(1+iu)}{\Gamma(1+(i-1/2)u)}\\ &\qquad\times\prod_{i=0}^{n-1}\frac{(1/2+iu/2)_{a+b}(1/2+iu/2)_b}{(1+(n+i-1)u/2)_{a+2b}}. \end{aligned}$$

The source reports the numerical evidence for $n=4,5$ and gives the conjectural values $P_1(x)=1$, $P_4(x)=1$, and $P_5(x)=(3+4x+8x^2)/15$, but does not provide a resolution status.