The tightness conjecture for antipodal n-fold covers of spheres

Let $Q(d,n)$ denote the minimum, over antipodal $n$-fold covers of the sphere $S^d$ by open sets, of the maximum number of covering sets containing any point. Theorem~ gives

$$\left\lceil \frac{d}{2}\right\rceil+n\leq Q(d,n)\leq d+n.$$

Tightness conjecture. For $n\geq 2$,

$$Q(d,n)=d+n.$$

The upper bound is attained by Gale's $n$-fold cover, and the conjecture asserts that this construction is optimal for every $n\geq 2$.