Conjecture on free infinite divisibility of the t-distribution

Let $\mathbf{t}_q$ denote the t-distribution with parameter $q$. A probability distribution is freely infinitely divisible (FID) if it is infinitely divisible with respect to free convolution, and $\mathcal{UI}$ denotes the stronger class mentioned in the source. The t-distribution conjecture. The t-distribution $\mathbf{t}_q$ is FID, and more strongly belongs to the class $\mathcal{UI}$, for every $q>\frac{1}{2}$.

The conjecture is motivated by the geometry of the component of $\{z\in\mathbb{C}:G_{\mathbf{t}_q}(z)\in\mathbb{C}^-\}$ containing $\mathbb{C}^+$ and by the preceding analytic verification on several parameter ranges. The source does not prove the assertion for all $q>\frac{1}{2}$.