The Poincaré-sphere surgery realization conjecture

Let $L(p,q)$ be a lens space, let $K\subset L(p,q)$ be a knot, let $g(K)$ denote its genus, and let $P^3$ denote the Poincaré sphere. An integer surgery is Dehn surgery with an integer framing. A Tange knot and a Berge knot of type VII refer to the knot classes described in the source. The Poincaré-sphere realization conjecture. Suppose that integer surgery along a knot $K\subset L(p,q)$ produces $P^3$. Then either $2g(K)-1<p$ and $K$ lies in the same homology class as a Tange knot, or $2g(K)-1=p$ and $K$ lies in the same homology class as a Berge knot of type VII. This is the analogue of the lens space realization problem for surgeries producing the Poincaré sphere. The proposed classification is supported by computations and partial results, but remains open; the source also notes the broader formulation for L-space homology spheres with $d$-invariant $-2$.