yes, when ms is turned on, it will do the IMSI ATTACH. The IMSI ATTACH is a normal Location Update procedure, only with one extra information that says “IMSI ATTACH”.
Basically, the MS does a LU with IMSI ATTACH. Therefore, as for a normal LU, VLR will update the LAI linked to that TMSI.
I don’t know why MS keeps the LAI after being switched off. It is perhaps used to speed up the cell selection. But it will NOT prevent the MS to do a Imsi Attach.
I believed that the information TMSI/LAI in the VLR would be erased when the MS does the IMSI DETACH. But you are suggesting that VLR keeps this information for a time = purge timer… Are you sure about it?