(I am a PhD student) I recently worked on a paper using a computational methodology developed by Professor X and my manuscript was published on a preprint server. My supervisor, Professor Y, is actively seeking collaborations and emailed Professor X to discuss potential opportunities. After reading my manuscript, Professor X was pleased and agreed to discuss further, but did not provide any help or contributions to the paper.
Professor Y then met with Professor X without me to discuss the potential for collaboration. Afterwards, Professor Y informed me that we should list Professor X as a co-author on the final version of my paper. I was not comfortable with this situation but Professor Y quickly shut that by stating that they had already discussed this with Professor X and everything was agreed.
After I emailed Professor X asking about their credentials to be included as a co-author on the paper. Professor X then replied to both Professor Y and me, clearly refusing the gift authorship.
I am wondering if I should email Professor X again to explain my perspective, as my reputation is at stake.
For more context about my circumstances, please refer to my two previous questions.