Hard to say - if the Tru Oil is a solvent for the dye, then it will happen every time... Tru oil contains mineral spirits, so if the dye is oil based then you have issues...
Read the label on the dye - and remember it is designed for leather, and check the cure times. But if it is transferring to your rag then I’d assume you would want a minimum 24 hours. If there are no time issues, give it a week. If Tru Oil pulls it up after that amount of time then tru oil is possibly not for this project