The prime-modulus minimum-color conjecture for links

Let $r>1$ be an integer, and let $L$ and $L'$ be links with nonzero determinants. For positive integers $a,b$, write $(a,b)$ for their greatest common divisor, and write $\langle a,b\rangle$ for $1$ when $(a,b)=1$, otherwise for their least common prime divisor. Let $mincol_r L$ denote the minimum number of colors modulo $r$ in a non-trivial Fox $r$-coloring of $L$, minimized over all diagrams of $L$. For each prime $p$, define

$$m_p:=mincol_rL$$

for any $L$ such that $p=\langle r,\det L\rangle$ and $\det L\ne 0$.

Prime-modulus minimum-color conjecture. If $\langle r,\det L\rangle=\langle r,\det L'\rangle$, then $mincol_rL=mincol_rL'$. Moreover, $m_p$ is a non-decreasing function of the prime $p$; in particular, $m_2=2$, $m_3=3$, and $m_5=m_7=4$.

The theorem preceding this statement establishes the displayed values for the primes $2,3,5,$ and $7$. The proposed extension concerns the behavior for all other primes and asserts that the minimum number of colors depends only on the least common prime divisor of $r$ and the determinant, while increasing monotonically with that prime.