Variation comparison conjecture for the F-distribution

Let $d_1\in\mathbb{N}$ and $d_2\in\mathbb{N}$ with $d_2\geq 5$. Let $X_{d_1,d_2}$ be an $F$-random variable with numerator and denominator degrees of freedom $d_1$ and $d_2$, respectively, and let $Z$ be a standard normal random variable. Variation comparison conjecture.

$$P\left\{|X_{d_1,d_2}-E[X_{d_1,d_2}]|\leq \sqrt{\operatorname{Var}(X_{d_1,d_2})}\right\}>P\{|Z|\leq 1\}\approx 0.6827,$$

and

$$\inf_{d_1,d_2}P\left\{|X_{d_1,d_2}-E[X_{d_1,d_2}]|\leq \sqrt{\operatorname{Var}(X_{d_1,d_2})}\right\}=P\{|Z|\leq 1\}.$$

The conjecture extends variation comparison inequalities previously established for many infinitely divisible continuous distributions. The paper proves the inequality for $d_1\in\{1,2,3,4\}$ and $d_2\geq 5$, while the general case and the infimum assertion remain to be established.