The sharp constant in the average absolute inner-product inequality

For each dimension $n$ and every $x\in\mathbb{R}^n$, define the average absolute inner product over the sign vectors $v_i\in\{-1,1\}^n$ by

$$\frac{1}{2^n}\sum_{v_i\in\{-1,1\}^n}|\langle x,v_i\rangle|.$$

The largest absolute constant $\alpha$ such that this quantity is at least $\alpha\lVert x\rVert_2$ for all $x\in\mathbb{R}^n$ and all dimensions $n$ is conjectured to satisfy

$$\frac{1}{2^n}\sum_{v_i\in\{-1,1\}^n}|\langle x,v_i\rangle|\geq\frac{1}{\sqrt{2}}\lVert x\rVert_2,$$

i.e. $\alpha=1/\sqrt{2}$. This identifies the proposed sharp universal constant in the average problem, which is used to improve the lower bound for the Banach–Mazur distance between the cube and the crosspolytope.