The realization conjecture for integer surgeries on lens spaces

Let $L(p,q)$ be a lens space, and suppose it is realized by integer surgery on a knot $K\subset S^3$.

Realization conjecture. Then $L(p,q)$ is realized by integer surgery on a Berge knot.

The source states that this conjecture would follow from the numerical conjecture above together with the theorem on simple knots and an argument using the Ozsváth–Szabó $d$-invariant. The supplied source gives no resolution status.