Edge-connectivity and pairwise disjoint perfect matchings in regular graphs

For $0 \leq t \leq r$ let $m(t,r)$ be the maximum number $s$ such that every $t$-edge-connected $r$-graph has $s$ pairwise disjoint perfect matchings. There are only a few values of $m(t,r)$ known, for instance $m(3,3)=m(4,r)=1$, and $m(t,r) \leq r-2$ for all $t \not = 5$, and $m(t,r) \leq r-3$ if $r$ is even. We prove that $m(2l,r) \leq 3l - 6$ for every $l \geq 3$ and $r \geq 2 l$.