Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?

Question

For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?

The Two Summands Conjecture states that surgery along a knot cannot produce a manifold with three or more prime factors. However, the best known upper bound for the number of prime factors is 3. I wonder whether there is a similar bound for surgeries along links with a given number of components.