Asking for recommendation letters for postdoc positions in mathematics
Things have changed since I was about to finish a PhD a long time ago, but the fact that you are unknown outside your university would have been common then, perhaps even now. 

What you suggest seems to me to have two downsides. First, you are putting all your eggs into one basket (the conjecture) and second, you are entering into a time consuming process where others will need to verify your work. But beyond that, those others don't really know much about you beyond this one thing. 

I would suggest, instead, that you depend on your advisor for a very strong letter and on others (professors and such) with whom you have interacted and who can speak now to your general mathematical ability and future potential. That can all be done quickly and can get you into what may be just the first round of applications. 

But don't abandon your stated idea here to reach out to those others who have seen your work and might also help with an application if the more common approach seems not to be working as well as you hope. 

You can get started early and if it doesn't work out, then the longer term solution may start to fall into place for the next round.