Monotonicity conjecture for spectral radii of token graphs

Let $G$ be a graph on $n$ vertices. For each $k$ with $1\le k\le \lfloor n/2\rfloor$, let $\rho_k(G)$ denote the spectral radius of its $k$-token graph. Monotonicity conjecture. For any graph $G$,

$$\rho_1(G)\le \rho_2(G)\le \cdots \le \rho_{\lfloor n/2\rfloor}(G).$$

The conjecture proposes that the spectral radii of the token graphs are nondecreasing with the number of tokens, complementing the analogous proposed monotonicity for their algebraic connectivities.