You say that both you and your ex-boyfriend are postgraduate students; if that is indeed the case then it would behoove you to behave in a manner that is a bit more mature than you have thus far displayed in your post. A "no contact" rule is, or should be, just that: No contact. You cannot decide for yourselves what is the more convenient way to observe what ought to be obvious; it mitigates the entire process and that's why you are at an impasse over what to do about the rumor mill involving him. Why should you do anything? If you are going to continue making a point of engaging yourself with what does not directly concern you, then what is the rhyme or reason of establishing a "no contact" rule? You and your ex-boyfriend have erected a rule that has no validity because neither of you sees fit to enforce it. Do the human math, if you will: He is your "ex," right? So that would ostensibly mean that neither you nor he are together, physically or in spirit; it is also supposed to mean that whatever is being said behind his back is no longer any of your business, no matter how friendly you feel toward him. You say that you'd made a pledge to inform him of any "drama" that is being exchanged beyond his earshot and that he also shares that obligation toward you. Once upon a time in our tortured lexicon, that was called "tattling" and it seems that despite the "no contact" rule that is supposed to exist between you and him, you both have given yourselves permission to be tattletales--to tell each other what others say about you.
For Pete's Sake, who cares?
As of 2017, words do not (yet) have the power to break bones, although they do retain the ability to mess up lives, sometimes to the Nth degree. Barring all that, you express hope that you will eventually patch things up with your ex but a little self-examination on your part may have to be the first requirement toward that end. First, if you hope to "fix" things then technically, your ex is not really your ex. Ex means forever but that is clearly not how you understand that prefix. Instead, it appears to mean "temporary separation"; a timeout is in fact what you have so let's call it what it is. You infer in your post that "I do want to Be single until we repair" and that, of course, is a decision you are free to make but you have not cleaved the umbilical cord that still attaches you to him; there is no "single" unless you are able to do that. The thing that you must bear in mind is that rumors do not begin in a vacuum, no matter how addle-brained; your ex-boyfriend is saying or doing something to incite them. Let him bear the responsibility of at least planting the seeds that feed the rumor mill and let him go handle it himself. You say that both you and he can "fend for ourselves" yet you still cling to each other; the second lesson to be learned in this impromptu teach-in about self-examination is this: Fending for yourself concerns only one person--you. You cannot stop people from saying what they will about him but you can stop babying him by worrying about the "drama" that swirls about him; that's his headache. Look to your own affairs and he, to his.