Rationality conjecture for Diophantine exponents of nilpotent Lie groups

Let $G$ be a connected nilpotent real Lie group endowed with a left-invariant geodesic metric, and let $\widehat{\beta}_k$ denote its Diophantine exponent on $k$ letters. Assume that $G$ is diophantine. Rationality conjecture. The limit

$$\lim_{k \to +\infty} \widehat{\beta}_k$$

exists and is a rational number. The rationality of this limit is known when the Lie algebra of $G$ is defined over $\mathbb{Q}$; the conjecture concerns the general Diophantine case without that rationality assumption.