Suppose we have two real monic polynomials \begin{align*} & p_1(t) = t^n + a_{n-1} t^{n-1} + \dots + a_0, \\ & p_2(t) = t^n + b_{n-1} t^{n-1} + \dots + b_0. \end{align*} Let $\alpha = (\alpha_1, \dots, \alpha_n) \subset \mathbb C$ and $\beta = (\beta_1, \dots, \beta_n) \subset \mathbb C$ be the roots of $p_1$ and $p_2$ respectively and assume no root in $\alpha$ and $\beta$ equal $0$. Necessarily, $\alpha$ and $\beta$ must be invariant under conjugation since the coefficients are real.
Does there exist nonzero real numbers $r_1, r_2$ such that the monic polynomials $\hat{p}_1(t)$ with roots $r_1 \alpha= (r_1 \alpha_1, \dots, r_1 \alpha_n)$ and $\hat{p}_2(t)$ with roots $r_2 \beta = (r_2 \beta_1, \dots, r_2 \beta_2)$ have either $a_{2j} = b_{2j}$ for $j =0, 1, \dots, [\frac{n}{2}]$ or $a_{2j+1} = b_{2j+1}$ for $j = 0, 1, \dots, [\frac{n-1}{2}]$ where $[s]$ denotes the greatest integer less than $s \in \mathbb R$?
Quadratic case seems always true. Is there a way to see the existence or nonexistence for higher orders?