I don't think I can share the log because I was playing on the beta, but suffice to say I bought my 7th Province on turn 15. I think this might be a legit combo.
The only things JoaT really lacks is +Buy and Copper trashing - Counterfeit provides both, all without the cost of an Action. CF can also put your Silvers on double duty and just keep your deck from getting too big in general. And then Jack just gives more Silvers for Counterfeit! I think the Copper trashing and Silver doubling might be more important than the +Buy, though.