  Let $F$ be a field of formal series over a finite field and $\mathcal{B}_d$
be the affine building associated to $PGL_d(F)$. Given a lattice $\Gamma$ in
$PGL_d(F)$, the complex arising as a quotient $\Gamma\backslash \mathcal{B}_d$
is called weakly Ramanujan if every non-tivial discrete simultaneous spectrum
of the colored adjacency operators $A_1,A_2,\ldots,A_{d-1}$ acting on
$L^2(\Gamma\backslash \mathcal{B}_d)$ is contained in the simultaneous spectrum
of those operators acting on $L^2(\mathcal{B})$. In this paper, we prove that
the standard non-uniform arithmetic quotient $PGL_4(\mathbb{F}_q[t])\backslash
\mathcal{B}_4$ of $PGL_4(F)$ is weakly Ramanujan.
