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: $P\{|X_{d_1,d_2}-E[X_{d_1,d_2}]|\le \sqrt{{\rm Var}(X_{d_1,d_2})}\}\ge P\{|W-E[W]|\le \sqrt{{\rm Var}(W)}\}$, 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 $5\le d_2\in\mathbb{N}$.