Rationality of the Euler-Chow series and finite generation of Cox rings for rationally connected varieties

Let $X$ be a smooth rationally connected projective variety of dimension $n$. The Euler-Chow series $E^1(X)$ is defined for divisors on $X$, and $[?]$.

Rationality–Cox ring conjecture. $E^1(X)$ is rational if and only if

$$\operatorname{Cox}(X)$$

is finitely generated.

This conjecture proposes that, for smooth rationally connected projective varieties, rationality of the Euler-Chow series is equivalent to finite generation of the Cox ring. The supplied text gives a counterexample to the corresponding question in general, while motivating this restricted class; no resolution is stated here.