Ooh... that's a good guess. I take back my Tacitus guess and go with @Finn235 ... Unless it's Tacitus, then I keep my original guess.
The overall look is very Balbinus, but looking at high-grade examples, they seem to be pretty uniform in showing a moustache, which the bust appears to lack. Both Tacitus and Gallienus often lack a moustache; but Tacitus is more corpulent, so my vote is:
That is a good point, but I was operating under the assumption that I thought there was only one or maybe two extant busts of emperor Tacitus. And while he can be rather porcine on his coins, he is markedly more thin and flabby in marble.
I'm going with the majority on this one. I think if the marble bust had a nose this one would be easier to identify .
Okay, it is time to tell you what the Palatine Museum label thought of bust number 9. I remind you that I was not confident that it could be firmly identified. But they, like @Finn235 , thought it was Balbinus. If so, it is interesting that a new emperor would have busts created so rapidly. Balbinus lasted only a little more than three months. They created wonderful coin portraits in that short time, but at least coins are functional and it is necessary for new emperors to have coins to pay the usual accession donatives. I imagine Balbinus was rather busy with the new administration and with trying to keep his head on his shoulders, but his staff somehow found time to make sure his statues were made. Balbinus, antoninianus, CONCORDIA AVGG. 23 mm. RIC 10. Balbinus, sestertius, PROVIDENTIA DEORVM 29 mm. RIC 19. I solicit more comments, but the ninth bust was the end of this group of portrait busts from the Palatine Museum in Rome. Thanks for participating!
It's also important to note that in 238, Balbinus was 60 years old, enormously wealthy, and had probably at least 35 years of civil service under his belt, including two consulships. I'd say it's likely that he already had a lot of marble busts long before he was hailed as Emperor. The "Balbinus Sarcophagus" for example most likely couldn't have been completed in just 3 short months and was probably started before 238:
Finn235, The Balbinus Sarcophagus is a masterpiece of Roman sculpture, something only the wealthy elite could afford. The Boston Museum of Fine Arts has a marble bust that they speculate could also be Balbinus, pictured below.