The integrability conjecture for quadratic harnesses

Let $(X_t)_{t\in(1-\delta,1+\delta)}$ be a quadratic harness on $(1-\delta,1+\delta)$ for some $\delta>0$, with parameters $(\eta,\theta,\sigma,\tau,\gamma)$. For a fixed $t$ in this interval, consider the moments of $X_t$. Integrability conjecture. (i) If $0<\sigma\tau<1$ and $1-2\sqrt{\sigma\tau}\leq\gamma\leq 1+2\sqrt{\sigma\tau}$, then

$$\mathbb{E}(|X_t|^p)<\infty$$

for all $0\leq p<2+\frac{1}{\sigma\tau}$. (ii) If $\sigma\tau$ is small enough and $-1\leq\gamma\leq 1-2\sqrt{\sigma\tau}$, then

$$\mathbb{E}(|X_t|^p)<\infty$$

for all $p\geq 0$. This describes the expected moment integrability of quadratic harnesses in parameter regimes determined by $\sigma\tau$ and $\gamma$; the supplied text does not indicate whether the claim has been proved or remains open.