Dyadic denominator law for the phase constants of the Jacobi zeros

Let $\kappa_r(A,B)\in\mathbb{Q}[A,B]$ be the $r$-th additive phase constant in the asymptotic phase equation for the zeros of Jacobi polynomials, with $A=\alpha^2$ and $B=\beta^2$. If $\operatorname{den}(\kappa_r)$ denotes the least positive integer clearing the denominators of all coefficients of $\kappa_r$, prove that, for every integer $r\geq 1$, $\nu_2(\operatorname{den}(\kappa_r))=E_r$, where $E_r=3r-1+\nu_2((r-1)!)$. Equivalently, $2^{E_r}\kappa_r\in\mathbb{Z}_2[A,B]$, while $2^{E_r-1}\kappa_r\notin\mathbb{Z}_2[A,B]$. The established odd-part bound is that the odd part of $\operatorname{den}(\kappa_r)$ divides $\operatorname{lcm}(1,3,\ldots,2r-1)$; the sharp conjecture concerns the exact dyadic exponent.