It has been conjectured, for any discrete density function $\{p_j\}$ on the integers, that there exists an $n_0$ such that the $n$-fold convolution $\{p_j\}^{\ast n}$ is unimodal for all $n \geq n_0$. A similar conjecture has been stated for continuous densities. We present several counterexamples to both of these conjectures. As a positive result, it is shown for a discrete density with a connected 3-point integer support that its $n$-fold convolution is fully unimodal for all sufficiently large $n$.