An analogue of Lehmer's conjecture for bounded-coefficient polynomial sequences

Let $N\geq 1$ be a natural number, and let $P_{2n+2l}$ be a sequence of integer polynomials defined in the source's equation (with coefficients bounded in modulus by $N$). Its limit ratio is the limiting ratio of the number of nonunimodular roots of $P_{2n+2l}(x)$ to its degree, when this limit exists. Analogue of Lehmer's conjecture. There is a constant $c(N)>0$ such that any such sequence whose limit ratio is strictly below $c(N)$ has limit ratio equal to $0$. The claim proposes a Lehmer-type gap near zero for limit ratios of sequences with uniformly bounded coefficients; the supplied text gives computational and theorem-based support but does not state that the claim has been proved or disproved.