If you are flying at constant CAS and the temperature changes the TAS will change but the MN will stay the same. For 'Doubting Thomas's' we can work through an example based on Q5299:
FL270/ -40ºC CAS 280kts: from CRP5: TAS = 412kts so MN = 0.69
FL270/ -50ºC CAS 280kts: from CRP5: TAS = 406kts so MN = 0.69
Mathematically: MN = TAS/LSS if temperature decreases then TAS and LSS both decrease keeping MN constant.
There are questions in the question bank covering all possible combinations; i.e.;
Temperature increase or decrease
CAS constant what happens to MN
MN constant what happens to CAS
The answer is "it remains constant".