A problem for Pauline authorship is the author’s reference to being restored to the congregation in Hebrews 13:19 and 23. The author and Timothy would even travel together to the congregation. There are two separate problems. The problem after 2 Timothy was written is that Paul reports that he has been condemned to die. This stopped all possible Pauline travel with anyone. But prior to 2 Timothy, Paul had commanded Timothy to remain in Ephesus (1 Tim 1:3) when Paul left Macedonia. Timothy was still in Ephesus when Paul wrote 2 Timothy. Paul and Timothy were not together. In fact, Timothy had not been imprisoned yet as stated in Hebrews 13:23.
This doesn’t argue in favor of Paul. It precludes him unless Hebrews was written before Paul left Timothy in Ephesus when Paul left for Macedonia about AD 63 (1 Tim. 1:3). During this time, Paul travels with Timothy around his evangelized cities and left Timothy in Ephesus, with Titus to Crete (Titus 1:5), spent the winter with Titus in Nicopolis (Titus 3:12) and evangelized the far west. This does not leave time to write Hebrews or travel with Timothy. During the period of Acts, Timothy was never imprisoned. There is no good time for Paul to write Hebrews and travel with Timothy after his imprisonment. This is an obstacle, not an argument in favor of Paul.