X gave some nanoparticles to Y lab 2 years ago. I worked on biological characterization of these particles in Y lab as a postdoc (second postdoc) and wrote a manuscript at the end of my 2 year contract. X lab had said when I joined Y lab that their work on the nanoparticles was almost ready for publication (so approximately 3 years) but till now they have not published the work.
Three years later, they haven't published the synthesis work. Now it is about a year since I left Y lab and there has been lack of communication across teams. Y lab people do not want to annoy X lab but do not have any plan in place of what is to be done. Y lab doesn't keep me in the loop and only when I ask they respond with a lame excuse for delay. If I question the excuse they ignore me till I send them a few follow-ups only with a new excuse.
Is there are any workaround the situation of the materials provided if I were to publish? And also if there is no technical reason, Y lab cannot keep blocking my paper from being put on the preprint server at least. So if at all they wash their hands off the paper (I don't think they will back off easily), how can I go about the issue of the materials provided that we used in our study if X lab refuses to be coauthors?