The X55 reflection-triangle concentric-circle conjecture

Let $T$ be a reference triangle, and let $T'$ be its $X_{55}$-reflection triangle. The six vertices of the P-hyperbolas of $T'$ passing through $X_{55}$ of $T$ are the six points associated with those hyperbolas. Concentric-circle conjecture. These six vertices lie on a circle concentric with the circumcircle of $T'$, and that circumcircle coincides with $X_7$ of $T$. The claim is presented as an experimentally supported “needle in a haystack” phenomenon; no proof or resolution is given in the supplied text.