Nonexistence conjecture for 8-regular K3-irregular graphs

Let $G$ be an $8$-regular graph. For a vertex $v$ of $G$, let its $K_3$-degree be the number of subgraphs isomorphic to $K_3$ that contain $v$. The graph $G$ is $K_3$-irregular if its vertices do not all have distinct $K_3$-degrees.

Nonexistence conjecture. There are no $8$-regular $K_3$-irregular graphs.

The conjecture would improve the bounds obtained in the paper, which show that any $8$-regular $K_3$-irregular graph would have order between $17$ and $22$. The preceding heuristic search found examples with exactly two equal pairs of $K_3$-degrees for orders $19$, $20$, $21$, and $22$, so the conjecture remains open based on the supplied text.