It will support the mattress, but these things should be over-engineered with a high
factor of safety.
Sometimes at parties and such, I've seen 3-4 kids sitting on the top bunk...you never know what the little monkeys
will come up with.
I think I would go with slats, overloaded webbing will cause a catastrophic failure that will be hard to repair and possibly quite dangerous. If the kids bounce on a slatted bed they might break a few slats but are unlikely to do any major damage and the slats are quickly and easily repaired.