There's a simpler solution for disposing (not storing) final nuclear waste: deep core drilling. You drill a 1m round hole 2km+ deep and dump shit there and fill back up. Unlike caves, natural or otherwise, on a properly selected site there is no possibility of things coming back up in millions of years. It's expensive, but not that expensive compared to the energy hitherto produced by the waste involved and eminently doable as it uses tech from the oil industry.

The main reason we don't do that is that most of the waste can be expected to be further used with future reactors.