  The $\mathrm{SU}(N)$ symmetric antiferromagnetic Heisenberg model with
multi-column representations on the two-dimensional square lattice is
investigated by quantum Monte Carlo simulations. For the representation of
Young diagram with two columns, we confirm that a valence-bond solid order
appears as soon as the N\'eel order disappears at $N = 10$ indicating no
intermediate phase. In the case of the representation with three columns, there
is no evidence for both of the N\'eel and the valence-bond solid ordering for
$N\ge 15$. This is actually consistent with the large-$N$ theory, which
predicts that the VBS state immediately follows the N\'eel state, because the
expected spontaneous order is too weak to be detected.
