This is a continuation of the previous question Comparison between first minimum of a lattice and a discrete subgroup in function field. Let $\mathbb{F}_q(T)$ denote the function field over $\mathbb{F}_q$, where $q$ is a prime power. The norm in this field is defined by
$ \left| \frac{f}{g} \right| = q^{\deg(f) - \deg(g)}, $ for $f, g \in \mathbb{F}_q[T^{-1}]$. Consider the field of Laurent series over $\mathbb{F}_q$, denoted by $\mathbb{F}_q((T^{-1}))$. It is well known that the set $ g \mathbb{F}_q[T]^d $ forms a lattice in the vector space $\mathbb{F}_q((T^{-1}))^d$, where $g \in SL(d, \mathbb{F}_q((T^{-1})))$.
Now, let us extend the field $\mathbb{F}_q((T^{-1}))$ by adjoining $T^{\frac{1}{2}}$, and denote this extension by $K$. In this setting, the ring extension $\mathbb{F}_q[T^{\frac{1}{2}}]$ forms a lattice in $K$, whereas $\mathbb{F}_q[T]$ does not. Given $\frac{f_1}{g_1}, \frac{f_2}{g_2} \in \mathbb{F}_q(T)^2$, we have $(f_1, f_2, g) \in \mathbb{F}_q[T]^3$.
Consider the flow $ g_t = \operatorname{diag}\left(T^{\frac{1}{2}}, T^{\frac{1}{2}}, T^{-1}\right) $ acting on $K^3$, and denote by $g_t \mathbb{F}_q[T^{\frac{1}{2}}]^3$ and $g_t \mathbb{F}_q[T]^3$ the corresponding lattice and discrete subgroup, respectively, in $K$. As observed in the previous discussion, the first successive minima coincide: $ \lambda_1\bigl(g_t \mathbb{F}_q[T^{\frac{1}{2}}]^3\bigr) = \lambda_1\bigl(g_t \mathbb{F}_q[T]^3\bigr). $ The question I pose is whether anything can be said about the $i$-th successive minima.
Since $\mathbb{F}_q[T] \subset \mathbb{F}_q[T^{\frac{1}{2}}]$, it follows that
$ \lambda_i\bigl(g_t \mathbb{F}_q[T^{\frac{1}{2}}]^3\bigr) \leq \lambda_i\bigl(g_t \mathbb{F}_q[T]^3\bigr). $ However, is it possible to make this inequality more explicit by identifying a constant, potentially depending on the flow or the degree of the field extension? Given that Minkowski's theorem does not hold for discrete subgroups in this context, I am uncertain whether such a comparison can be established.
