In this article:

Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503.

the author defines a ring $R$ to have the unique generator property on right ideals if for all pairs $a,b\in R$, $aR=bR$ implies $a=bu$ for some unit $u\in R$. (In other articles it is just abbreviated to UGP. In fact the definition seems to go back all the way to Kaplansky.)

The author comments that they do not know if the condition is left-right symmetric. In my brief search, I was unable to unearth any resources declaring that this had been decided one way or the other since then.

Is anyone aware if it is now known if UGP is or isn't symmetric?

Update :

Also in

Khurana, Dinesh, and T. Y. Lam. "Rings with internal cancellation." Journal of Algebra 284.1 (2005): 203-235.

the authors there remark that it is unknown at that time.