Variation comparison between the $F$-distribution and the normal distribution

Let $X_{d_1,d_2}$ be an $F$-random variable with numerator and denominator degrees of freedom $d_1$ and $d_2$, respectively. We investigate the inequality: $P\{\mathrm{\mid }{X}_{d_1,d_2}-E[X_{d_1,d_2}]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X_{d_1,d_2})}\}\ge P\{\mathrm{\mid }W-E[W]\mathrm{\mid }\le \sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(W)}\}$, where $W$ is a standard normal random variable or a ${\chi }^2(d_1)$ random variable. We prove that this inequality holds for $d_1\in \{1,2,3,4\}$ and $5\le d_2\in \mathbb{N}$.