Symmetric-matrix extension of the quadratic theorem for totally real algebraic integers

Work over $\mathbb{C}$. For a totally real algebraic integer $\lambda$ of degree $d>1$, let $E_s(n,\lambda)$ be the maximum multiplicity of $\lambda$ as an eigenvalue of an $n$-vertex graph. Symmetric-matrix quadratic conjecture. The extension of Theorem 3.3 to symmetric matrices holds for totally real algebraic integers; in particular, for $d>1$,

$$E_s(n,\lambda)\geq n/d-o(n).$$

The paper proves this assertion for degrees $d\leq4$ and for all representable $\lambda$, where representability means that some integral symmetric matrix has eigenvalues exactly $\lambda$ and its conjugates. The general case is left open in the source.