Prism manifold realization conjecture for positive surgery parameters

Let $p>1$ and $q>0$ be relatively prime integers. Let $P(p,q)$ be the prism manifold associated with these parameters, let $K$ be a knot in $S^3$, and let $\mathcal P^+$ denote the prism manifolds with $q>0$ arising from surgeries on knots in $S^3$. Prism manifold realization conjecture. If $P(p,q)$ can be obtained by $4q$--Dehn surgery on $K$, then $P(p,q)\in\mathcal P^+$. The theorem preceding this conjecture classifies the corresponding negative-$q$ cases; the positive-$q$ integer surgery realization problem remains open, and this conjecture proposes its solution.