The singleton Minkowski spectrum conjecture for function-field lattices

Let $\mathcal{L}_d$ be the space of function-field lattices and define the Minkowski spectrum by

$$\mathcal{S}_d=\{\mu(\Lambda):\Lambda\in\mathcal{L}_d\}.$$

Here $q$ is the parameter appearing in the function-field absolute value. Singleton Minkowski spectrum conjecture. For every $d\geq 2$,

$$\mathcal{S}_d=\{q^{-d}\}.$$

The source presents this as a conjecture based on known results, including the exact statement in dimension $2$, the general upper bound, and computer calculations; no general proof is supplied.