thermodynamics - Does "IF isentropic THEN reversible" only holds for adiabatic processes?

I know that: IF adiabatic and reversible THEN isentropic

First question: does the implication IF isentropic THEN reversible hold for adiabatic processes?

Second Question: if yes to the above, are there other processes other than adiabatic for which it is true?

Answer

Thank you to everyone who contributed to this thread. Now, at least the first question, has an answer.

As clearly described at this related question, the answer to the first question is: IF adiabatic and isentropic THEN reversible.

In other words, for adiabatic processes, "isentropic" and "reversible" are equivalent notions.

As an aside, I notice here that it is also possible to arrive at this conclusion using the approach to adiabatic processes of Lieb and Yngvasson, for which the entropy is the only functional that completely characterizes the existence of an adiabatic process between two thermodynamical states.

But the second question remains, namely:

Does "IF isentropic THEN reversible" only hold for adiabatic processes?