Before revision:   Finally, we disprove a conjecture of Goldrup and Phillipson, and then present an example of a code that is neither open convex nor closed convex.
After revision:  Additionally, while adding non-maximal codewords can only increase the open embedding dimension by 1, here we demonstrate that adding a single such codeword can increase the closed embedding dimension by an arbitrarily large amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and then present an example of a code that is neither open convex nor closed convex.