Last Played

Question regarding unlocking achievements for games I no longer own and have not played for a while.

So I have a period of internet connection down time coming up, so I will be working on offline achievements on the 360 for a while.

I want to unlock some offline achievements on games I no longer own, games that have like one or two achievements left for me too get, but obviously these games are quite far down on my list as I haven’t played them for a while.

So my question is, how does Horizon work, can I move the games up my recently played list, or will unlocking achievements move them up the list anyway?

I don’t want to unlock the last achievement I need on MW2, only for the game to stay half way down my games list when it sync with live when my internet comes back on, that just raises suspicion surely.