Let W be the Weyl a group of a semisimple simply connected group over C.

Let I={1,...,r} the set of simple roots.

For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple reflexions that appear in a reduced decomposition of w.

Let w an element such that supp(w)=I and length(w)>r+1, is it true that there exist an element w' such that

1/ $w'\leq w$

2/length (w')=r+1

3/supp(w')=I

?