The Nielsen–Borsuk–Ulam dichotomy conjecture for free involutions on tori

Let ${\mathbb{T}}^n$ be the $n$-torus, let $f:{\mathbb{T}}^n\to{\mathbb{T}}^n$ be a map, and let $\tau$ be a free involution of ${\mathbb{T}}^n$. Nielsen–Borsuk–Ulam dichotomy conjecture. The Nielsen–Borsuk–Ulam number satisfies

$$\operatorname{NBU}(f,\tau)\in\{2^{n-1},0\}.$$

The conjecture is motivated by the computations for free involutions on low-dimensional tori and by the example yielding $\operatorname{NBU}(g,\tau_2)=2^{n-1}$. It predicts that for every map and free involution the invariant can take only these two values; the supplied source does not state a resolution.