Transcendence of conjugacy growth series for Baumslag–Solitar groups

For an integer $k\geq 2$, let $BS(1,k)=\langle a,t\mid tat^{-1}=a^k\rangle$, and let a finite generating set mean any finite generating set of this group. The Baumslag–Solitar transcendence conjecture. The conjugacy growth series of $BS(1,k)$ with respect to any generating set are transcendental. The paper proves the corresponding statement for the standard generating set $\\{a,t\\}$; extending transcendence to arbitrary generating sets remains open.