Nonexistence conjecture for lens space knots in oppositely oriented Brieskorn spheres

Let $\Sigma(p,q,r)$ be a Brieskorn homology sphere with its usual orientation, and let $-\Sigma(p,q,r)$ denote the same manifold with the opposite orientation. Let $K_{p,k}$ denote a parameterized lens space knot. Nonexistence conjecture. The oppositely oriented Brieskorn homology sphere $-\Sigma(p,q,r)$ contains no $K_{p,k}$. The source says this is an analogue of a previously cited result and that it can be checked for the examples in the paper, but does not state that it has been proved in general.