Before we can answer this question, we need to discern the method that the author of the Pastoral Epistles used to compose similar passages. I propose that the author had access to some of Paul's letters and used them to create verisimilitude, but made errors by consulting only one text at a time.
Historical errors made by the author
2 Tim 4:19 indicates that the author believed that Prisca and Aquila were in Ephesus, and he would have inferred that from 1 Cor 16:19. But it is clear from Rom 16:3 that Prisca and Aquila had moved to Rome before Paul's imprisonment there.
A further problem is created by the statement in 2 Tim 4:20 that Paul left Trophimus ill in Miletus. Acts 20:15 shows that Paul did indeed stop at Miletus, but Acts 21:29 shows that he did not leave Trophimus there.
1 Tim 1:3 says that Paul left Timothy in Ephesus while there were adversaries there, and went himself to Macedonia. This situation is a natural inference from 1 Cor 16:5-11, where Paul expects Timothy to be with him in Ephesus when he departs for Macedonia, and where there are adversaries. The problem is that 2 Cor 1:1 shows that Timothy did not stay in Ephesus.
1 Tim 1:3; 3:14, Titus 1:5; 3:12 require that Paul visited Ephesus and the Aegean after an imprisonment in Rome. However, Acts 20:25, which should be trusted, says that Paul knew that he would not be able to return to Ephesus. It would be too dangerous for Paul to go back there after he illegally delivered the collection.
1 Tim 3:12 says that Timothy was young. The author may have inferred Timothy's youth from 1 Cor 4:17, where Paul calls Timothy his child (likewise Phil 2:22). The problem is that, while Timothy was probably younger than Paul, he could not have been young at any time that 1 Timothy could have been written.
2 Timothy 1:5 suggests that Timothy was a third generation Christian, but this is historically unlikely since his conversion must have been before 49 CE.
2 Tim 4:9-18
It might be suggested that this passage, unlike other parts of the PE is somehow more authentic. This is special pleading. Moreover, there are indications that the author made the same types of errors in this passage that he made elsewhere.
He refers to Lucius-Luke and Demas by their diminutive name forms, rather than by the long forms of their names (Lucius and Demetrius). I can see no reason why Paul would use the diminutive forms here, and note that 2 Tim 4:19 uses the name form "Prisca", which found in Paul's letters, rather than the diminutive, "Priscilla". In Philemon Paul had good reason to use diminutive name forms, but the author of Colossians simply copied the names unaltered. Likewise the author of 2 Tim 4:10-11 seems to have copied these names from Philemon or Colossians.
Similarly, Philemon 23 mentions the name "Mark", which was probably the praenomen of someone who would ordinarily have been called by his cognomen. The author of Colossians seems to have misunderstood this and wrongly identifies him as John-Mark. The author of 2 Tim 4:11 may have taken the name "Mark" from Philemon or Colossians, for we have no evidence that Paul had a co-worker who normally went by that name.
2 Cor 4:13 seems to have been inspired by 2 Cor 2:12-13. The author could easily have imagined that Paul forgot his coat when he left the Troad in a great anxiety. The problem is that Acts 20:5 shows that Paul would have had opportunity to pick up his coat if he had left it in Troas. Some suppose that Paul deliberately left his coat, knowing that he was going to Jerusalem, but Jerusalem and Rome have very similar temperatures.
More importantly, for me, the author makes a big blunder by using the name "Titus". The two brothers of 2 Cor 8:18-22 and the brother of 2 Cor 12:18 are given protective anonymity because of their role in delivering and organizing the collection. Paul, when referring to Timothy's collection work, calls him by his lesser-known name, "Titus", in 2 Corinthians to afford him similar protection. By this time "Timothy" was the name that Paul generally used for him. The mention of "Titus" in 2 Tim 4:10 is therefore out of place, particularly in a letter that was addressed to Timothy! The author simply copied the name "Titus", perhaps from 2 Cor 2:12-13, which he seems to have known (see above).
Finally, 2 Tim 4: seems dependent on Ephesians 6:21.
The combined weight of these points makes me feel confident that 2 Tim 4:9-18 was written by the author of the Pastoral Epistles, who had little historical information about Paul's companions.