$\require{AMScd}$ I'm considering minimum values (at non-zero integer points) of real, positive-definite, quadratic forms of determinant $1$. These are functions $f:\mathbb{R}^n\to \mathbb{R}$ which can be represented as $x\mapsto x^\intercal Ax$ for some symmetric, positive-definite matrix $A$ with determinant $1$. Call this function space $\mathcal{Q}_n$.

Since I'm interested in extreme values (at integer points), I'm going to identify two such functions $f \sim f'$ if there is some invertible integer matrix $\Gamma$ such that $f'(\cdot)=f(\Gamma \cdot)$. This gives me a set $\mathcal{Q}_n/\sim$.

Now (using Hermite or Minkowski) one can easily demonstrate uniform (over $f$) bounds such as $$\min\limits_{v\in \mathbb{Z}^n\smallsetminus 0}f(v) \leq \left(\frac{4}{3}\right)^{(n-1)/2}, $$ so that $$S_n:=\sup\limits_{f} \left\lbrace\min\limits_{v\in \mathbb{Z}^n\smallsetminus0}f(v)\right\rbrace $$ is finite. Moreover, one can show (by a compactness argument due to Mahler) that in each dimension $n$, this bound is achieved by some particular quadratic form.

**Question:** In each dimension $n$, is there a finite bound on the equivalence classes $[f] \in \mathcal{Q}_n/\sim$ that can attain this maximum? That is, can we have an infinite number of inequivalent $f$ for which
$$\inf\limits_{v\in \mathbb{Z}^n\smallsetminus0}f(v) = S_n?
$$

For example, in dimension $2$ (resp. $3$) Hermite (resp. Gauss) showed that this max is only attained by (an appropriate multiple of) $[x^2+xy+y^2]$ (resp. $[x^2+y^2+z^2+xy+xz+yz]$).

Thanks for reading! Sorry if this question is not appropriate.