I think the answer is always yes. Considering the rational canonical
form, one can deform a matrix continuously into a nilpotent matrix
while maintining the sizes of the constituent companion matrices.
Now keep the largest companion matrix fixed and scale down the remaining
towards zero. We get a continuous path from our starting matrix to a
conjugate $A$ to the direct sum $B_m$ of the zero matrix and the companion matrix
of $x^m$. If the conjugacy is by a matrix of positive determinant, then it can
be done via a path. But that is sufficient as there is a negative determinant
matrix commuting with $B_m$.