We have seen this many times where the light will not dim or go out, but they were charging fine.
I take it you bought a used GT as they haven't been made for several years and if it was new than the battery been sitting around for a while and in that case may have to be charged up and used a few times for it to get fully charged. If it was used when you got it you should ask the previous owner if this did it for him and to check it is the factory charger as I have seen some use other charger not for the GT.
On the Elite and the GT they use a circuit board inside the pack to regulate the current and when it get charged it will slow down charging and may even quit charging and this is when I believe the light will go dim or shut off, so if it is not charging the battery which you will find out if it only last a few minutes of use, then you know it is not charging, but if it is I would just use it as it is working fine.
I wonder how many that have Elites or GT that the light don't go off or dim when done charging?? Mine I use the Sun Ray SR12 charger which connect right to the clips on the battery so mine don't go though the charging port so I don't know if mine dims or go off, but the Elite and the GT I had before I know the light never went off with charging though the charge port, but they did charge with no problem.
Rick