I also use a cube (60) and I reckon there is a bit of noise gating going on in some of the models at low volumes.
This cuts the sound when the input level drops below a certain level, killing the sustain. This seems to go away when you crank it but can be a problem at living room levels. You can get round this by boosting the input a tad. I use an old Marshall Shredmaster set just to boost the level for this and it works great.
Have a listen to my contribution to the little wing collab. to see how it sounds:
"Little Wing" Jam Collab
That`s a humbucker on a fat strat too.
I imagine that a compressor pedal will work just as well.
Will