I've taken an oximeter with me on flights before, and can give a data point of n=1 to say this isn't the case.
My oxygen % decreased initially with altitude until it stabilised around 90-92% (at ~10k feet, IIRC). It sat at this point for the whole flight (8 hours), with a low of 87%.
Through forced rapid deep breaths I could get it back up into the mid-high 90s, but also got light headed and funny looks.
Moving around the cabin would also raise it.