Saturation property fails for Schubert coefficients

The saturation property for Littlewood–Richardson coefficients was established by Knutson and Tao in 1999. In 2004, Kirillov conjectured that the saturation property extends to Schubert coefficients. We disprove this conjecture in a strong form, by showing that it fails for a large family of instances. We also refute the saturation property for Schubert coefficients under bit scaling and discuss computational complexity implications.