Slave societies exist as long as those who benefit from slavery hold significant political power. Slave societies are far worse in creating overall wealth than free societies.
For a slave society to remain, you need;
A. Powerful patricians with influence, large tracts of land or mines, industry etc.
B. The lack of a large self-owning class of peasants.
C. The lack of a large class of proletarians.
D. Victorious wars to create a steady influx of slaves.
Now, it is not entirely impossible that this system could be broken.
An Emperor, faced with a patrician revolt, siezes patrician land and distributes it among the army, creating a class of self-owning peasants.
Add a law the eldest son inherits everything, and you get a good class of free men that look for work, a countryside proletariat to work at wineries, in industries, mines etc.
Add a few slave rebellions, some Pax Romana (not bringing in new slaves), perhaps some of the well-educated Greek slaves writing plays and books of slavery and the misery of it and the class of self-owning peasants becoming a political factor (they could demand the end of slavery, as it would be of no beneift to them, and absolutely of no benefit to their proletariat second and third sons) and an Imperial Decree banning slavery could very well happen.
It will take time, but it is not impossible and will not need ASB technology introduction.