On Plotkin's Question on Two-Variable Identities Characterizing Finite Soluble Groups: Two Infinite Families of PSL(2,p) and Finite Certificates

B. I. Plotkin asked in the Kourovka Notebook (Problem 15.75 b), 2002) whether a finite group is soluble if and only if it satisfies one of the identities u_n = 1, where u_1 = [x,y] and uₙ₊₁ = [[u_n,x],[u_n,y]]. We give a partial answer: the question reduces to the minimal simple groups; PSL(2,p) is settled for all primes p ≡ ±1 (mod 16) and, in the Notebook's commutator convention, for all primes p ≡ 0, ±1 (mod 7), and every minimal simple group of order less than |PSL(2,10007)| has a pair whose sequence never reaches 1; the question stays open. The reduction to Thompson's list of minimal simple groups is standard: it was outlined for this sequence by Nikolova and Plotkin (2001) and carried out by Bandman, Greuel, Grunewald, Kunyavskii, Pfister and Plotkin for another sequence. The families are explicit: if x, y ∈ SL(2,p), tr x = 0 and tr y = tr xy is a root of 2t^4 − 4t^2 + 1, then u_3 = −u_2 ≠ ±1 (in the other commutator convention u_4 = u_3 ≠ ±1); if tr x = 1 and tr y = tr xy is a root of t^3 ± t^2 − 2t ∓ 1, then u_4 = u_2 ≠ ±1 (in the Notebook's convention). By Dirichlet's theorem they contain half of the primes p > 3 with p ≡ ±2 (mod 5), for which PSL(2,p) is minimal simple. Checked finite certificates, pairs with u_M = uM+L ≠ 1, settle PSL(3,3), the 626 groups PSL(2,q) of Thompson's list with q ≤ 10^4, and Sz(8), Sz(32), Sz(128); for PSL(3,3) and PSL(2,p), 5 ≤ p < 100, Nikolova and Plotkin had reported such pairs, found by computer. Not decided are the other groups of Thompson's list: PSL(2,p) for primes p > 10^4 outside the families, PSL(2,2^r) for primes r ≥ 17, and PSL(2,3^r) and Sz(2^r) for primes r ≥ 11. This is computational progress and two infinite families, not a near-solution. No printed answer to the question 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-15.75 (Kourovka Notebook, Problem 15.75 b), B. I. Plotkin, 2002).