Alon-Jaeger-Tarsi conjecture

The Alon-Jaeger-Tarsi conjecture states the following:

For any (finite) field F with at least four elements and any non-singular matrix M over F there is a vector x such that both x and Mx have only non-zero entries.

The problem is easy to state – one can understand it after attending any Linear Algebra I lecture course in Germany. In the case that the number of elements of F is a proper prime power, the conjecture was solved by Alon and Tarsi (link) and at first glance it seems to me to be an elementary proof. Now interestingly, the case of the number of elements of F being a prime power remained open until recently: It was resolved only last year by Nagy and Pach (arXiv:2107.03956), but again interestingly only in the case that the field has more than 61 (and not 79) elements.