Kolmogorov's converse exponential inequality for lower capacities

Let $\{X_{n,i};i=1,\ldots,k_n\}$ be an array of independent random variables in the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with

$$\underline{s}_n^2=\sum_{i=1}^{k_n}\widehat{\mathcal E}[X_{n,i}^2].$$

Let $x_n$ be a sequence of positive numbers with $x_n\to\infty$ and $x_n/\underline{s}_n\to0$. Suppose

$$\frac{\sum_{i=1}^{k_n}|\widehat{\mathbb E}[X_{n,i}]|}{\underline{s}_nx_n}\to0,\qquad \frac{\sum_{i=1}^{k_n}|\widehat{\mathcal E}[X_{n,i}]|}{\underline{s}_nx_n}\to0$$

and there exists a positive number $\alpha$ such that $|X_{n,i}|\le\alpha\underline{s}_n/x_n$ for $i=1,\ldots,k_n$. Kolmogorov's converse exponential inequality. We conjecture that, for every $\gamma>0$, there exists a positive constant $\pi(\gamma)$ such that

$$\liminf_{n\to\infty}x_n^{-2}\ln\mathcal V\left(\sum_{i=1}^{k_n}X_{n,i}\ge z\underline{s}_nx_n\right)\ge-\frac{z^2}{2}(1+\gamma)$$

for all $0<z\alpha\le\pi(\gamma)$. This would provide the lower-capacity analogue of Kolmogorov's converse exponential inequality in the sub-linear expectation setting; the stated result is presented as a conjectural extension of the corresponding upper-capacity estimate.