Distinct spectral radii for Type 5 digraphs

Let $n\in\mathbb{N}$, $5\leq n$, and let $r_k,s_k,t_k$ satisfy

$$4\leq r_k\leq s_k\leq t_k<n,\qquad r_k+s_k+t_k=2n,$$

for $k=1,2$, with $(r_1,s_1,t_1)\neq(r_2,s_2,t_2)$. For each $k$, let $\rho_k$ be the only real positive root of

$$x^n-x^{n-r_k}-x^{n-s_k}-x^{n-t_k}-2.$$

Distinct-root conjecture. One has $\rho_1\neq\rho_2$. This asserts that distinct admissible parameter triples determine distinct spectral radii for the corresponding Type $5$ digraphs. The claim was computationally verified for $5\leq n\leq 200$; the conjecture proposes that it holds for every $n\geq5$.