Stability of Szarek's inequality with best constant

Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be independent Rademacher random variables. We settle the best constant $c$ such that $$\mathbb{E}\mid \sum _{j=1}^n{a}_j{\epsilon }_j\mid \ge \frac{1}{\sqrt{2}}+c\mid a-\frac{e_1+e_2}{\sqrt{2}}\mid$$ holds for every $n\ge 2$ and every unit vector $a\in {\mathbb{R}}^n$ with $a_1\ge \cdots \ge a_n\ge 0$. We prove that the optimal constant is $c=(2-\sqrt{2}{)}^{3\mathrm{/}2}\mathrm{/}8$, with equality attained at $a=\frac{1}{2}(1,1,1,1,0,\dots ,0)$ for $n\ge 4$. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.