Classification conjecture for equal determinants of colored paths

Let $P$) and $Q$ be two colored paths on $m$ vertices with concentration matrices $K_P$ and $K_Q$ satisfying

$$\det(K_P)=\det(K_Q).$$

A reflection of a colored path is the path obtained by reversing its vertex order. Colored-path determinant conjecture. If $m$ is even, then one of the following holds: $Q$ is identical to $P$; $Q$ is a reflection of $P$; or $P$ and $Q$ satisfy the color configuration stated in the even-path theorem, or a reflection of that configuration. If $m$ is odd, then one of the following holds: $Q$ is identical to $P$; $Q$ is a reflection of $P$; or $P$ and $Q$ satisfy the color configuration stated in the odd-path theorem, or a reflection of that configuration. The claim is presented as an unresolved conjecture based on the necessary color-multiset conditions proved earlier.