How to proceed when the baseline (state-of-the-art) published results claim much better performance than I can reproduce?
There is absolutely no reason that you can't publish a paper that says "We compared our method to methods X and Y. Since code the original code was not available for X and Y, we reimplemented the methods to the best of our ability. The code for these reimplementations is available in supplementary files A and B. Our new method out performed the reimplementations of X and Y by z%. However, it should be noted that it was not possible to reproduce the reported results for X and Y. "

People who want to know will have to look at your re-implementations and decide themselves if they think you have correctly re-implemented. 

Seniority has nothing to do with it - be transparent, and the world will judge if they believe you or the people that won't release their code.