Inspired by this quesion. I'm curious about the maximal lower bound of $|a_n|$, where $n\geq 1$ and $a_n$ is the coefficient of $x^n$ in the symmetric-coefficient polynomial
$$ P(x) = x^{2n}+a_1x^{2n-1}+ a_2 x^{2n-2}+ ...+ a_{n-1}x^{n+1}+a_nx^n+ a_{n-1}x^{n-1}+...+a_2 x^2+a_1x+1 $$
with all real roots (not necessarily distinct). The short answer of @abacaba in the link there shows $|a_n| \geq 2$, but this is not tight. By examining closely, for $n=1,2,3,4$ the maximal lower bounds are $2,2,4,6$, and correspond to the polynomials $(x-1)^2, (x^2-1)^2, (x^2-1)^2(x-1)^2, (x^2-1)^4$, respectively. I want to have a tighter bound in this post but don't know any simple way for the following conjecture:
Conjecture: The maximal lower bound of $|a_n|$ is the coefficient of $x^n$ of the polynomial $ (x-1)^{n}(x+1)^{n}$ if $n$ is even and $(x-1)^{n+1}(x+1)^{n-1}$ if $n$ is odd.