  In Secion~1 we describe what is known of the extent to which a separable
extension of unital associative rings is a Frobenius extension. A problem of
this kind is suggested by asking if three algebraic axioms for finite Jones
index subfactors are dependent. In Section~2 the problem in the title is
formulated in terms of separable bimodules.
  In Section~3 we specialize the problem to ring extensions, noting that a
biseparable extension is a two-sided finitely generated projective, split,
separable extension. Some reductions of the problem are discussed and solutions
in special cases are provided. In Section~4 various examples are provided of
projective separable extensions that are neither finitely generated nor
Frobenius and which give obstructions to weakening the hypotheses of the
question in the title. We show in Section~5 that existing characterizations of
the separable extensions among the Frobenius extensions in are special cases of
a result for adjoint functors.
