Powers of sign portraits of real matrices

The sign portrait $S$ of a real $n\times n$ matrix is a matrix over the semiring with elements $0,1,-1$ and $\theta$, where $\theta$ symbolizes indeterminateness. It is proved that if $k$ is the least positive integer such that all the entries of $S^k$ are equal to $\theta$ then $k\le2n^2-3n+2$, and this bound is sharp.