Today the SCOTUS was to confer on the Hollister rehearing/petition. I don’t know what other viable cases are still pending. Seems they’ve about run their course, so this might be the last shot.
snip...
The courts are not the only entities empowered to deal with such a question. A committee of the House of Representatives could be authorized to conduct an investigation into the eligibility issue. The classic formulation of the Congressional role is Woodrow Wilson's, in his 1884 book Congressional Government: