Mostly true, but not absolutely:

https://en.wikipedia.org/wiki/Varronian_chronology

> The Varronian chronology was adopted by the Roman state during the first century BC and gave rise to the traditional years ab urbe condita ("from the founding of the city"); most especially, those dates were used in monumental Augustan-era inscriptions, the fasti Capitolini and the fasti Triumphales.[40]