ink will never cure if it hasn't when it comes off the printer. latex doesn't work that way...
a heat gun will do the trick, but getting the profile working is obviously the way to go before you start printing tons of vinyl.
Where do you get that from? I've seen latex (and SEPIAX) prints that were completely wet to the touch when they came off, and were completely cured a few days later just by sitting at ambient temperatures.
I've also seen dripping wet solvent prints cure after sitting out for a while.
Even normal water based inks dry/cure at room temperature.
So where is this coming from? Half of the issue with latex ink is that you need heat to make the water and co-solvents in the ink evaporate, so that it doesn't drip off the material. The other half is the polymerization of the "latex" resins, which occurs at room temperature but is greatly accelerated by heat.
Evaporation can be blocked by an overlaminate, but if the ink isn't running, then all of the water and co-solvents have already evaporated. A greasy feel indicates that the polymerization of the resins hasn't finished. You don't need air to complete that process, only time (and heat, if you want to speed it up).