Generic lens spaces are spectrally unique

For positive integers $n$ and $q$, let $\mathfrak L(n,q)$ be the isometry classes of $(2n-1)$-dimensional lens spaces with fundamental group of order $q$. Define the spectrally unique subset

$$\mathfrak L^{\bullet}(n,q)=\\{L\in\mathfrak L(n,q):\operatorname{Spec}(L)\neq\operatorname{Spec}(L')\quad\forall L'\neq L\text{ in }\mathfrak L(n,q)\\}.$$

For $x>0$, set

\mathcal U_n(x)=\frac{\sum_{q\leq x}\\#\mathfrak L^{\bullet}(n,q)}{\sum_{q\leq x}\\#\mathfrak L(n,q)}.

Generic lens-space spectral uniqueness conjecture.

$$\lim_{x\to\infty}\mathcal U_n(x)=1$$

for all $n\in\mathbb N$. This is a proposed extension of Wolpert's generic spectral-uniqueness results for Riemann surfaces and flat tori. Since the space of lens spaces has no natural topology, the conjecture uses density among spaces whose fundamental-group order is at most $x$; the source mentions numerical evidence but does not state a proof.