I wonder if my employer is lying and vaguely threatening me.

My employer has made a statement "Time to complete these training sessions can be during Teacher Work Day (8/26/22), throughout the month of Sept during your prep times and/or at your own time and preference since this is part of our professional obligation of working in the education field. " which i believe is a lie since the training is mandatory and under FLSA should be paid training and employees are under no obligation to complete the training on their own time unless otherwise stated in a contract. This mandatory training must also be competed by custodians, secretaries, food service workers and other hourly/salaried positions with no previous contractual agreements to complete training and with no time allotted for them to do so during the work day.