Abstract. Thomassen [ J. Combin. Theory Ser. B, 141 (2020), pp. 343–351] asked whether every [Formula: see text]-edge-connected [Formula: see text]-regular graph of even order has [Formula: see text] pairwise disjoint perfect matchings. We show that this is not the case if [Formula: see text]. Together with a recent result of Mattiolo and Steffen [ J. Graph Theory, 99 (2022), pp. 107–116] this solves Thomassen’s problem for all even [Formula: see text]. It turns out that our methods are limited to the even case of Thomassen’s problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan–Raspaud conjecture, the Berge–Fulkerson conjecture, and the 5-cycle double cover conjecture.