On Brandl's Question on Finite Simple Groups of Bounded Engel Depth: A Reduction to the Suzuki Groups

For r ≥ 1 let V_r be the class of finite groups G that satisfy a law [x, _r y] = [x, _s y] for some s = s(G) > r. R. Brandl asked (Kourovka Notebook, Problem 11.16 b)) whether V_r contains only finitely many nonabelian simple groups. Brandl proved that V_2 contains none (1983), and Brandl and Nikolova proved that for q ≥ 4 the group PSL(2,q) lies in V_3 only for q ∈ {4, 5, 8} (1986). We give a partial answer to part b): for every r, all but finitely many nonabelian finite simple groups in V_r are Suzuki groups, so b) holds if and only if the Engel depth of Sz(q) tends to infinity; this last statement and part a) stay open. Here the Engel depth of G is the least r with G ∈ V_r; part a), on the Fitting length of soluble groups in V_r, is not treated. Extending the method of Brandl and Nikolova, we prove that PSL(2,q) ∉ V_r for all q > Q_r, with an explicit Q_r: the Fricke trace recursion turns long Engel chains into rational points of an absolutely irreducible plane curve, which the Weil bound provides. Alternating, classical and exceptional groups other than Sz(q) have alternating sections of unbounded degree or sections PSL(2,q), and the classification of finite simple groups completes the reduction. For an odd prime p, every proper section of Sz(2^p) has Engel depth at most 3, so no argument through sections can reach these groups. As evidence only, we compute the Engel depths 62 of Sz(8) and 527 of Sz(32), and lower bounds from explicit pairs up to 792,500 for Sz(8192). No printed answer to part b) was found in the sources searched; we claim no priority. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-11.16 (Kourovka Notebook, Problem 11.16, R. Brandl, 11th issue, 1990).