On the length of global integrals for $G{L}_n$

In this paper we prove Conjecture \ref{conj1} for a set of representations of the group $G{L}_n(\mathbf{A})$. This Conjecture is stated in complete generality as Conjecture 1 in \cite{G2}, and here we prove it for various cases. See Conjecture \ref{conj2} below. First we prove it in the case when the length of the integral is four, and then we discuss the general case.