The torus-knot invariant formula conjecture

Let $T(p,q)$ be a torus knot of type $(p,q)$, where $p$ and $q$ are coprime integers. Let $\rho_{j,k}\colon \pi T(p,q)\to\operatorname{SO}(3)$ send the standard generators $a$ and $b$ to rotations through angles $2\pi j/p$ and $2\pi k/q$ about distinct axes, respectively, with $1\leq j\leq\lfloor p/2\rfloor$ and $1\leq k\leq\lfloor q/2\rfloor$. Denote the corresponding invariant by $I_{j,k}(T(p,q))$. The torus-knot invariant formula conjecture.

$$I_{j,k}(T(p,q))=\frac{1}{(pq)^2}\left(4\sin\frac{j\pi}{p}\sin\frac{k\pi}{q}\right)^4.$$

This is the paper's proposed formula for the Euclidean invariant associated with the specified nonabelian representation of a torus knot. It is presented as an observed expression rather than proved generally, so its validity beyond the calculations discussed remains open.