Generalized potential-density conjecture for varieties with non-nef anticanonical divisor

Let $V$ be a smooth variety defined over a number field, and suppose that the divisor $-K_V$ is not numerically effective. Assume there is no unramified finite morphism

$$f:U\to V$$

such that there is a dominant rational map

$$g:U\dasharrow Z,$$

where $Z$ is a variety of general type with $\operatorname{dim}(Z)>0$.

Generalized potential-density conjecture. Under these assumptions, rational points on $V$ are potentially dense.

This generalizes the preceding conjecture and is motivated by examples relating potential density to the absence of maps to positive-dimensional varieties of general type. The source gives no general proof; it notes that the two potential-density conjectures are logically negations of the weak Lang conjecture.