Salve, Amici!
When C. Suetonius lists the evidence of Caesar's power abuse (abusus dominatione ) on Ch. LXXVI sec. I of his Divus Iulius, he specifically states that the Dictator accepted the title Imperator as part of his name (insuper praenomen Imperatoris); ie, for him (and presumably for his family), the use of the title (and the triumphal condition) was going to be permanent, the same as the consulship (continuum consulatum) and the dictatorship (perpetuam dictaturam ):
"Non enim honores modo nimios recepit: continuum consulatum, perpetuam dictaturam praefecturamque morum, insuper praenomen Imperatoris, cognomen Patris patriae, statuam inter reges, suggestum in orchestra;"
"For not only did he accept excessive honours, such as an uninterrupted consulship, the dictatorship for life, and the censorship of public morals, as well as the forename Imperator, the surname of Father of his Country, a statue among those of the kings, and a raised couch in the orchestra;"