Upper-bound conjecture for disjoint perfect matchings in edge-connected r-graphs

For $0\leq t\leq r$, let $m(t,r)$ be the maximum integer $s$ such that every $t$-edge-connected $r$-graph has $s$ pairwise disjoint perfect matchings. The upper-bound conjecture. For all integers $l\geq 2$ and $r\geq 2l$, one has

$$m(2l,r)\leq l-1.$$

The paper has already proved the weaker upper bound $m(2l,r)\leq 3l-6$ for $l\geq 3$ and $r\geq 2l$; this conjecture would substantially sharpen it.