Asymptotic cosine conjecture for elliptic Faulhaber polynomials

Let $\mathcal{F}_m(\lambda)$ denote the elliptic Faulhaber polynomial and let $\mathcal{B}_{2m-2}$ denote the corresponding elliptic Bernoulli number. For real $x$ in a finite interval, asymptotic cosine conjecture. As $m$ tends to infinity,

$$\frac{\mathcal{F}_m\left(\frac{x^2+x}{2}\right)}{2\mathcal{B}_{2m-2}}\longrightarrow\frac{1-\cos 2\pi x}{2\pi^2}.$$

The claim is known in the hyperbolic limit $g_2=g_3=0$, which gives the usual Bernoulli polynomials. Its proposed justification uses the expected convergence of the Lamé density-of-states asymptotic expansion at integer $x=n$, arising from finite-gapness, but the elliptic statement remains unproved.