The new density conjecture for symmetric pairs of roots

Let $f(x)$ be a polynomial of degree $n$. For a prime $p$ in $Spl(f)$, number the roots of $f(x)$ modulo $p$ as

$$0\le r_1\le\dots\le r_n<p.$$

For a subset $S\subseteq\{1,2,\dots,n\}$, write $Pr(f,S)$ for the limiting frequency table of the quantities $\left\lceil\sum_{i\in S}r_i/p\right\rceil$ over splitting primes, when this limit exists. The new density conjecture. Suppose that $f(x)$ is not equal to $g(h(x))$ for any quadratic polynomial $h(x)$. Then, for every $j$ with $1\le j\le n$, one has

$$Pr(f,\{j,n+1-j\})=[1/2,1/2].$$

Here $[1,1]/2$ denotes $[1/2,1/2]$. The statement is presented as a conjecture based on numerical experiments; it predicts an equal limiting frequency for the two possible values of the sum of each symmetric pair of normalized roots, subject to the stated non-composition hypothesis.