Böröczky's polyhedral norm conjecture

Let $u_1,\dots,u_{2d}$ be unit vectors in ${\mathbb R}^d$, and define

$$P=\bigcap_{i=1}^{2d}\{x\in{\mathbb R}^d:\langle u_i,x\rangle\leq 1\}.$$

Böröczky's conjecture. There is a point $x\in P$ with norm

$$\lVert x\rVert=\sqrt d.$$

This conjecture is closely related to the paper's upper bound for absolute convex hulls and is cited there in a different formulation. The supplied text gives no resolution, so its status remains open.