Kirby–Zhang surgery conjecture for the Poincaré homology sphere

Let $K$ be a knot in $S^3$. Suppose there exists a rational number $r$ such that the $3$-manifold obtained by $r$-surgery on $K$ is homeomorphic to the Poincaré homology sphere $\Sigma(2,3,5)$. Kirby–Zhang's surgery conjecture. Then $K$ is the left-handed trefoil knot. This conjecture concerns the possible rational surgeries yielding the Poincaré homology sphere; the paper states it as a conjecture and derives the relevant genus-one consequence, while the general assertion remains open in the supplied text.