The Euler-line concurrence classification for circumcircle centers

Let $ABC$ be a triangle, let the Euler line be its line through the circumcenter and orthocenter, and let $X_{99}$ be the specified center of the triangle. Two centers are in the same orbit when they are related by the equivalence used for triangle centers in the source.

Euler-line concurrence conjecture. A center on the circumcircle is concurrent for the Euler line if and only if it lies in the orbit of $X_{99}$.

The preceding results establish concurrence for several individual centers, including $X_{131}$, $X_{136}$, $X_{403}$, and $X_{974}$, by computer-algebra verification. The stated classification of all circumcircle centers is left as a question for further investigation.