  Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B,
141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph
of even order has $r-2$ pairwise disjoint perfect matchings. We show that this
is not the case if $r \equiv 2 \text{ mod } 4$. Together with a recent result
of Mattiolo and Steffen [Highly edge-connected regular graphs without large
factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves
Thomassen's problem for all even $r$. 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.
