The answer is yes, but but you will not notice any difference.
Container is "rigid" but it will have a slightly changing volume in relation to the relative pressure difference between the two walls – in and outside the sealed unit. But it is designed to run at such high volumes that the increase in size of the chamber is completely minimal. Otherwise the chamber would flex bigger as your pumping it up and this would lower pressure causing you to pump more and more and then bang.
It is a bit like you holding your breath and going underwater – you are a sealed unit – but not rigid, you will decrease in volume – air in lungs under higher pressure. Chamber just designed to minimise this "desire" to change volumes to equalise pressure.
I would lump it in with.
a) Will a hot day make my tyres feel different?
b) Will Helium in my tyres make my wheels have less mass?
c) Will Helium stay in my tyres?
d) Will I ride more grrrr with facial hair on my rigid SS.