Algebraicity conjecture for perturbative invariants of fiber knots

Let $M$ be a Seifert fibered space with orbifold Euler characteristic $\chi>0$, let $\sigma$ be the geometric parameter of the Seifert space, and let $X\in\{A,B,C,D\}$ denote a series of Lie algebras, with $BCD$ interpreted as the combined series appearing in the source. Set $L$ to be a finite-degree extension of $\mathbb{Q}(e^{-\chi u_0/(2\sigma)})$.

Algebraicity conjecture. The extension $L$ can be chosen depending only on the ambient manifold and the Lie-algebra series $X\in\{A,BCD\}$, and, at least for fiber knots, the perturbative invariants in any representation are Laurent expansions at $u_0\to0$ of elements of $L$.

This conjecture extends the explicitly computed algebraic-function description for particular Seifert spaces and Lie types; the source gives no general proof or resolution.