The dismissive partners tend to drive their mate toward attachment anxiety, by failing to respond well or at all to reasonable messages that are requesting reassurance. This dismissive behavior tends to mold even the most secure partner to gradually change toward more anxious and insecure. If the dismissive partner recognizes the problem and takes some responsibility for trying to respond positively even when he doesn't really feel like it, this can gradually change the dismissive partner and better the relationship.
On the other hand, if the dismissive partner seems to purposively not communicate, as a form of the mental game, then talking openly and eventually ending the relationship is something that should be considered. If time goes by and you see that communication is not the right tool to remedy the relationship, the self-confident partner would know that someone better can be found and would not be too afraid to give up on the bad and negative relationship.