Finiteness conjecture for Seifert surgeries with fixed hyperbolic base orbifold

Fix a $2$-orbifold $S^2(k,l,m)$ with $k,l,m>1$. Consider Dehn surgeries on hyperbolic knots in $S^3$ that produce Seifert fibred spaces having $S^2(k,l,m)$ as their base orbifold. Finiteness conjecture for fixed base orbifolds. For every fixed $2$-orbifold $S^2(k,l,m)$, with all $k,l,m$ larger than $1$, there are only finitely many slopes with which Dehn surgeries on hyperbolic knots in $S^3$ can produce Seifert fibred spaces with $S^2(k,l,m)$ as the base orbifold. The theorem immediately preceding the conjecture proves finiteness for certain fixed orbifolds and the surrounding results establish further special cases; the conjecture proposes this finiteness for every such fixed base orbifold.