Euler-transform coefficient conjecture for the partial theta function

Let $\Theta_0(x,y)=\sum_{n=0}^{\infty}x^n y^{n(n-1)/2}$ be the partial theta function, and let $\xi_0(y)$ be its associated formal power series. Write

$$\xi_0(y)=\prod_{m=1}^{\infty}(1-y^m)^{-c_m}.$$

Euler-transform coefficient conjecture. The sequence $(c_m)_{m=1}^{\infty}$ is strictly positive, increasing, strictly convex, and satisfies $\Delta^k c\geq0$ for $k=3,4$; explicitly, these conditions are $c_m>0$, $\Delta c\geq0$, $\Delta^2c>0$, and $\Delta^k c\geq0$ for $k=3,4$. The sequence $\Delta^5c$ begins with $-3$, so the conjecture does not extend to $k=5$. The claims were verified through order $y^{6999}$, but no proof was known.