  We construct highly edge-connected $r$-regular graph which do not contain
$r-2$ pairwise disjoint perfect matchings. The results partially answer a
question stated by Thomassen [Factorizing regular graphs, J. Comb. Theory Ser.
B (2019), https://doi.org/10.1016/j.jctb.2019.05.002 (article in press)].
