Can’t tab into Popper from material UI split button demo

I’m following the demo of the split button here But it looks like the demo itself isn’t accessible (you can’t tab from the button to the popper).

I’ve tried swapping/adding aria-controls for aria-owns but it’s not letting me move between the button and popper with a keyboard only.

How can I make this accessible? I’m presuming theres a good reason the demo doesn’t Menu (from a brief look, I don’t think it’s as easy to position perhaps??