I am looking for an asymptotic expansion of J(n)

$J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$ The first approximation is managed to get $F_1(n)=\frac {n!} {\sqrt{\pi A}}$, $A=n(n-1)(2n+5)/36$

*Is a general expansion known for this?*