Exact spectral threshold conjecture for perfect matchings in balanced 3-partite 3-graphs

Let $H$ be a $3$-partite $3$-graph with three vertex classes of size $n$. For each vertex $v$, let $N_H(v)$ be its link graph and let $\rho(N_H(v))$ denote the spectral radius of its adjacency matrix. Define

$$\tau(n)=\begin{cases} \displaystyle\sqrt{\frac{n(n-1)}{2}},&\text{if $n$ is odd},\\[12pt] \displaystyle\frac12\sqrt{n^2-n+2+\sqrt{n^4-6n^3+9n^2+12n-12}},&\text{if $n$ is even}. \end{cases}$$

Exact spectral threshold conjecture. If

$$\rho(N_H(v))>\tau(n)$$

for every vertex $v\in V(H)$, then $H$ contains a perfect matching.

The conjecture proposes the exact finite-$n$ strengthening of the paper's asymptotic spectral theorem. The displayed odd and even constructions show that the threshold is asymptotically tight, but the supplied source does not state that this exact form has been proved or refuted.