On Part b) of Kourovka Notebook Problem 19.93: Finite p-Groups with Unbounded Lower p-Central Factors and No C_p ≀ C_p Quotient for p = 2, 3, 5

Problem 19.93 of the Kourovka Notebook, posed by P. Spiga in 2018, conjectures that there is a function f: ℕ × ℕ → ℕ such that every factor of the lower p-central series of a finite p-group with d generators and without an epimorphic image isomorphic to C_p ≀ C_p has order at most f(p,d); part a) concerns p ≥ 7, part b) p = 2, 3, 5. M. Kinyon and J. Urban refuted part a), in a preprint that declares AI assistance, with a filtered Golod–Shafarevich inequality and a counting bound for truncated group algebras; the method of this note is theirs. We show that part b) is false: for each of p = 2, 3, 5 no bound f(p,d) exists (for p = 5 with any d ≥ 3, for p = 3 with any d ≥ 5, for p = 2 with any d ≥ 7). What is new is the choice of relators: x_i^p and (x_i x_j)^p on three generators for p = 5 and on five for p = 3, where the argument of Kinyon and Urban in C_p ≀ C_p, applied to each pair of generators, excludes this quotient; and twelve products of squares and commutators on seven generators for p = 2, where the exclusion of C_2 ≀ C_2 becomes a finite statement about twelve bilinear forms over 𝔽_2, checked exactly in two ways. We also observe why the Golod–Shafarevich argument with the relations used for p ≥ 7 fails for p = 2, 3, 5. Every step is proved in the note, including the Golod–Shafarevich inequality in the form used. With part a), the conjecture fails for every prime. The cases (p,d) = (5,2), (3,2), (3,3), (3,4) and (2,2), …, (2,6) are not decided. No printed answer to part b) was found in the sources searched. 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-19.93 (Kourovka Notebook, Problem 19.93, P. Spiga, 19th issue, 2018).