Medvedev–Scanlon conjecture on dense orbits of rational difference varieties

Let $k$ be an algebraically closed field, let $V$ be a variety over $k$, and let $(V,\phi)$ be a $\sigma$-variety over $k$, with $\phi$ the self-map defining its difference structure. Say that $(V,\phi)$ is $\sigma$-rational when it has no nonconstant $\phi$-invariant rational functions. A $k$-point has Zariski-dense orbit if the Zariski closure of its orbit under $\phi$ is all of $V$. Medvedev–Scanlon conjecture. If $(V,\phi)$ is $\sigma$-rational then there exists a $k$-point with Zariski-dense orbit.

The conjecture is an arithmetic-dynamical analogue of the relationship between rationality and dense leaves in differential algebra. It is known when $k$ is uncountable by a theorem of Amerik and Campana, but remains open for countable algebraically closed fields.