You do need to be aware that HDPE is permeable to oxygen, which is very different to whether it is airtight for our purposes or not. A number of wine producers use it to bulk store and age wine for that reason. However, I don't think it is as simple as saying HDPE is permeable therefore it's bad, more that there is a risk that should be understood if you're going to make the best use of it. Even oxidation is not necessarily bad in the right beer
Personally, I've never had an issue with storing wort, fermenting it and then storing finished beer in HDPE cubes. although the storing would generally only be for weeks, I have lagered or aged beer in a cube a few times for longer waiting for a keg to be free, without apparent ill effect. Back in the UK collapsible polypins were (and presumably are) regularly used for cask ales at a Homebrew level and by some breweries, without (to my knowledge) complaint of oxidation. I'd suspect they are far more permeable to oxygen than HDPE cubes and that the CO2 dissolved in the beer and the active yeast help.
So, based on experience I'd have to say you should be fine lagering in HDPE for a few months, but at some point beyond you might encounter issues.
Or what manticle said