The limiting alternative-measure conjecture for $P_{m,1}(x)$

Let $P_{m,1}(x)$ be the polynomial family defined earlier in the paper, and let $LC(P_{m,1}(x))$ denote its limit ratio. Corollary $\mathrm{CorSeq}$ bounds every adherent point $A$ of the sequence by

$$\frac{2}{\pi^2}\leq A\leq\frac{2}{3\pi}.$$

Limiting alternative-measure conjecture. The sequence has a limit, and

$$\lim_{m\to\infty}LC(P_{m,1}(x))\approx 0.209.$$

The stated value lies between the lower and upper limiting bounds from the preceding corollary, but the candidate supplies numerical evidence rather than a proof.