A graphical group whose number of irreducible characters of degree p^3 is not a polynomial in p

For a finite simple graph Γ and an odd prime p, let G_Γ(F_p) be the largest group of class at most two and exponent dividing p generated by elements x_v, one for each vertex v, such that x_u and x_v commute whenever u and v are not adjacent. Rossmann asked (Oberwolfach Report 38/2021; Question 1.10 in Bull. London Math. Soc. 2022) how the number of irreducible characters of G_Γ(F_p) of degree p^i depends on p, and suspected that these functions might be wild. We show that they are not always polynomials. Let Γ_MK be the bipartite graph on the points and lines of the Möbius–Kantor configuration 8_3 in which a point is joined to the five lines not containing it. For every odd prime p, G_{Γ_MK}(F_p) has exactly p^10 (h(p) + (1 + (−3/p)) (p − 1)^15) irreducible characters of degree p^3, where h is an explicit integer polynomial of degree 15 obtained by an exact computation; the count is a polynomial on each residue class modulo 3, but on no set of primes containing infinitely many primes ≡ 1 and infinitely many ≡ 2 (mod 3), and this needs no computer. We use the description of character degrees by ranks due to O'Brien and Voll and the fact, classical over the complex numbers, that 8_3 can be realised exactly over the fields containing a root of t^2 − t + 1. The counts are polynomials for i = 1, and for i ≤ 2 if Γ is bipartite. Conditionally on the Mnëv–Sturmfels theorem as stated by Belkale and Brosnan, who sketch its proof, some count is not even a polynomial on residue classes (existence only). The general behaviour remains open. 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: OWR-8415341-012 (Rossmann, problem session, Oberwolfach Report 38/2021, Computational Group Theory, pp. 2081–2082; Question 1.10 of Bull. London Math. Soc. 54 (2022)).