This is a really old question but this is still the first one that shows up in google so answering it here. This conjecture is now solved. Here’s a quantamag post on it

 https://www.quantamagazine.org/long-sought-math-proof-unlocks-more-mysterious-modular-forms-20230309/

This is a really old question but this is still the first one that shows up if you search for "unbounded denominators conjecture" so answering it here. This conjecture is now solved.

Here's the arXiv preprint from Frank Calegari, Vesselin Dimitrov, Yunqing Tang https://arxiv.org/abs/2109.09040 (Thanks @michael-albanese)

Here’s a quantamag post on it  https://www.quantamagazine.org/long-sought-math-proof-unlocks-more-mysterious-modular-forms-20230309/