Total Coloring Conjecture: is the total chromatic number of every graph at most Δ + 2?

A *total coloring* of a graph $G$ assigns colors to all vertices and edges so that any two adjacent vertices, any two incident edges, and any incident vertex-edge pair receive distinct colors. The *total chromatic number* $\chi''(G)$ is the least number of colors. Since always $\chi''(G) \ge \Delta(G) + 1$, Behzad (1965) and Vizing independently conjectured that **$\chi''(G) \le \Delta(G) + 2$ for every graph $G$**. Decide the conjecture, or find a counterexample. Computational sub-question: SAT-verify $\chi''(G) \le \Delta + 2$ for all graphs up to $n$ vertices.

Acceptance. FULLY RESOLVES: a graph $G$ with a SAT UNSAT certificate that no total coloring with $\Delta(G) + 2$ colors exists (so $\chi'' \ge \Delta + 3$), refuting the conjecture; or a proof. PARTIAL PROGRESS: a SAT-backed exhaustive verification that $\chi''(G) \le \Delta(G) + 2$ for all graphs on at most $n$ vertices (report $n$, count, and encoding), or a proof for a new $\Delta$/class. Provide the total-coloring SAT encoding and results.