Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables with the probability density function
\[
f_X(x) = \begin{cases}
\frac{1}{\sqrt{2 \Gamma\left( \frac{1}{6} \right)}} x^{-\frac{5}{6}} e^{-\frac{x}{8}}, & 0<x<\infty, \\
0, & \text{elsewhere},
\end{cases}
\]
where \( \Gamma(\cdot) \) denotes the gamma function. Also, let \( \bar{X}_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n) \). If
\[
\sqrt{n} \left( \bar{X}_n - \mathbb{E}[\bar{X}_n] \right) \xrightarrow{d} N(0, \sigma^2),
\]
then \( \sigma^2 \) (rounded off to two decimal places) is equal to ________