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 \(f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}\) 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 \(C_p\wr C_p\) has order at most \(f(p,d)\); part a) concerns \(p\ge7\), 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 \(d\ge3\), for \(p=3\) with any \(d\ge5\), for \(p=2\) with any \(d\ge7\)). What is new is the choice of relators: \(x_i^p\) and \((x_ix_j)^p\) on three generators for \(p=5\) and on five for \(p=3\), where the argument of Kinyon and Urban in \(C_p\wr C_p\), 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 \(C_2\wr C_2\) becomes a finite statement about twelve bilinear forms over \(\mathbb{F}_2\), checked exactly in two ways. We also observe why the Golod–Shafarevich argument with the relations used for \(p\ge7\) 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 \((2,2),\dots,(2,6)\) are not decided. No printed answer to part b) was found in the sources searched. This is an unrefereed note.