Positivity conjecture for elliptic Bernoulli numbers

Let $\mathcal{B}_{2m}$ be the elliptic Bernoulli numbers, and let $w$, $\eta$, $g_2$, and $g_3$ denote the associated Weierstrass quantities. Positivity conjecture. The elliptic Bernoulli numbers have the form

$$\mathcal{B}_{2m}=(-1)^{m-1}\bigl(\hat A^{(m)}(g_2,g_3)w-\hat B^{(m)}(g_2,g_3)\eta\bigr),$$

where the polynomials $\hat A^{(m)}(g_2,g_3)$ and $\hat B^{(m)}(g_2,g_3)$ have positive rational coefficients. This is proposed as a potentially easier route to proving the alternating-sign property for the reduced elliptic Faulhaber polynomials. No general proof is supplied, although the claim is a stated conjecture.