The classification conjecture for surgery in the kernel of a colouring

Let $p$ be an odd prime, let $(K,\rho)$ be a $p$-coloured knot, and consider equivalence classes of $p$-coloured knots under $\pm1$-framed surgery along loops in $\ker\rho$, called surgery in $\ker\rho$. The invariant of the paper is defined on these equivalence classes.

Classification conjecture. The invariant is a complete invariant of the set of equivalence classes modulo surgery in $\ker\rho$. Equivalently, the theorem asserting that there are exactly $p$ equivalence classes, represented by connect-sums of $n$ left-hand $(p,2)$-torus knots with a given colouring for $n=1,2,\ldots,p$, holds for every odd prime $p$.

The paper proves this classification for $p=3$ and $p=5$; the conjecture extends the result to all odd primes. The proposed invariant is intended to distinguish the $p$ representative classes.