The shirts left perfect and the complaint arrived a month later. That timing is the clue, and it points at the dryer rather than at the press.
What dye migration is
Polyester is coloured with disperse dye, which turns to gas at cure temperatures.
The gas passes into the ink film sitting on top of it and colours it from beneath.
Why it shows up late
The reaction continues gently after cure, especially in a warm carton.
So a perfect shipment becomes a pink one somewhere between the dryer and the shelf.
How it is prevented
A low-bleed underbase blocks the migrating dye before it reaches the top colour.
A lower cure temperature with a longer dwell reaches full cure with less sublimation.
What to ask for
Whether a low-bleed underbase is being used on every dark blend, as standard.
And a wash-and-warm-storage test on the actual garment, not on a cotton sample.