Conjecture on the eigenvalue interval of the transposition graph

Let $S_n$ be the symmetric group on $[n]:=\{1,\dots,n\}$, let $T_n$ be the set of transpositions in $S_n$, and let $Cay(S_n,T_n)$ be the Cayley graph of $S_n$ generated by $T_n$, with two vertices $f,g\in S_n$ adjacent if and only if $fg^{-1}\in T_n$. Its eigenvalues are the eigenvalues of its adjacency matrix. Eigenvalue interval conjecture. There exists a constant $n_0$ such that for $n\ge n_0$, all integers in the interval

$$\left[-{\lfloor (2n+1)/3\rfloor\choose 2},\;{\lfloor (2n+1)/3\rfloor\choose 2}\right]$$

are eigenvalues of $Cay(S_n,T_n)$. Earlier results establish consecutive eigenvalue intervals of linear and quadratic length, but leave a gap between them; this conjecture asserts that the entire symmetric interval up to the quadratic bound is eventually contained in the spectrum.