  We use the equivariant $\mu$-bubbles technique to prove that for any compact
manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant
of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is
positive. This generalized the $S^1$-stability conjecture of Rosenberg to
compact manifolds with boundary.
