Conjecture on the eigenvalues of the Laplacian on $C^i(X^n_\varnothing)$

Let $q$ be the parameter in the construction of $X^n_\varnothing$, let $0\leq i\leq n-1$, and let $\Delta$ act on $C^i(X^n_\varnothing)$. The positive eigenvalues of $\Delta$ are allowed to depend on $q$, as is the dimension of $C^i(X^n_\varnothing)$. Eigenvalue conjecture. The number of distinct eigenvalues of $\Delta$ depends only on $i$, not on $q$. Moreover, as $q\to\infty$, the positive eigenvalues of $\Delta$ tend to the integers

$$n-i,\ n-i+1,\ \dots,\ n+1.$$

The examples for $n=1,2$ and additional calculations for $n=3$ motivate the claim, but no proof or resolution is supplied here; in particular, the asserted independence from $q$ and the limiting description remain open.