  Bound secret information is classical information that contains secrecy but
from which secrecy cannot be extracted. The existence of bound secrecy has been
conjectured but is currently unproven, and in this work we provide analytical
and numerical evidence for its existence. Specifically, we consider two-way
post-processing protocols in prepare-and-measure quantum key distribution based
on the well-known six-state signal states. In terms of the quantum bit-error
rate $Q$ of the classical data, such protocols currently exist for
$Q<\frac{5-\sqrt{5}}{10}\approx 27.6\%$. On the other hand, for
$Q\geq\frac{1}{3}$ no such protocol can exist as the observed data is
compatible with an intercept-resend attack. This leaves the interesting
question of whether successful protocols exist in the interval
$\frac{5-\sqrt{5}}{10}\leq Q<\frac{1}{3}$.
  Previous work has shown that a necessary condition for the existence of
two-way post-processing protocols for distilling secret key is breaking the
symmetric extendability of the underlying quantum state shared by Alice and
Bob. Using this result, it has been proven that symmetric extendability can be
broken up to the $27.6\%$ lower bound using the advantage distillation
protocol. In this work, we first show that to break symmetric extendability it
is sufficient to consider a generalized form of advantage distillation
consisting of one round of post-selection by Bob on a block of his data. We
then provide evidence that such generalized protocols cannot break symmetric
extendability beyond $27.6\%$. We thus have evidence to believe that $27.6\%$
is an upper bound on two-way post-processing and that the interval
$\frac{5-\sqrt{5}}{10}\leq Q<\frac{1}{3}$ is a domain of bound secrecy.
