Common slots of bilinear and quadratic Pfister forms

We say that I^n(F) is m-linked if any m bilinear n-fold Pfsiter forms have a common (n-1)-fold factor. In a recent publication, Karim Becher pointed out that when F is a global field, I^n(F) is m-linked for every positive integer m, and raised the question of whether I^n(F) being 3-linked implies that it is m-linked for every positive integer m. In the special case of characteristic 2, this question can be phrased in two versions - one for bilinear forms and another for quadratic forms. We will provide negative answers to both versions of the question in characteristic 2 and discuss some open problems.