Multiplicity one theorem over characteristic 2

It is shown for all local fields $\mathbb{F}$ which are of characteristic different from $2$ that any distribution on $GL_{n+1}(\mathbb{F})$ which is invariant under conjugation by $GL_n(\mathbb{F})$ is also invariant under transposition. In this paper we give an adaptation of the proof of this theorem to fields of characteristic 2.