Assume $X=\mathbb{C}\mathbb{P}^{n}$, $\omega$ is the Fubini-Study Kähler form and $B_{R}$ is the euclidean ball centered at the origin and of radius $R$ in a chart $\mathbb{C}^{n}\subset\mathbb{C}\mathbb{P}^{n}$. We have explicitly computed the extremal function in this case (example 3.5). This yields

$$T_{\omega}(B_{R})=\frac{R}{\sqrt{1+R^{2}}}.$$

Observe that $T_{\omega}(B_{R})\sim R$ as $R\rightarrow 0$. This shows the optimality of the rate of decreasing in proposition 3.8.