The simplest answer is, "What are your flying patterns?" If you normally only fly one carrier and that works fine for you, then there's no reason to get status on another one. Doing the RTW on your preferred carrier will have the advantage of making it that much easier to maintain (or increase) your status with them, plus you'll get all the bonus miles, better treatment, etc from having status with the airline you'll be spending so much time on.

However, if you find yourself splitting your flying time naturally between more than one airline, then it may make sense to do the RTW on the second airline. I guess the main question is, what benefit will getting status on the second airline get you? status on an airline you never fly isn't really worth all that much...

Dan