The Esperet-Hendrey-Lagoutte-Marseloo-Norin-Steiner (2025) flow reconfiguration problem has a positive answer

For a finite multigraph G and an integer k ≥ 2, let Mk(G) be the graph on the nowhere-zero ℤk-flows of G, and Ik(G) the graph on the integer flows F of G with 1 ≤ |F(e)| ≤ k − 1 on every edge; two flows are adjacent if the support of their difference is a cycle. We prove that Mk(G) and Ik(G) are connected for every finite bridgeless loopless multigraph G if k = 40, if k = 48, or if k is a multiple of 4 with k ≥ 52. This gives a positive answer to Problem 1.2 of Esperet, Hendrey, Lagoutte, Marseloo, Norin and Steiner. The same holds for k = 66, for every prime power pN with N ≥ 6, and for a set of moduli of natural density one; these results do not cover every k ≥ 40. For one fixed graph G it holds for every k above a bound that depends on G, primes included. The main tool is a master theorem for a finite abelian group Γ with a subgroup B and a set of steps. It gives the connectivity of the flow graph F(G, Γ) for every finite abelian group Γ that is not cyclic of prime-power order and has a subgroup of order at least 4 and index at least 10, for example every abelian group of order 40.