Incompressibility of generic torsors of norm tori

Let $p$ be a prime integer, $F$ be a field of characteristic not $p$, $T$ the norm torus of a degree $p^n$ extension field of $F$, and $E$ a $T$-torsor over $F$ such that the degree of each closed point on $E$ is divisible by $p^n$ (a generic $T$-torsor has this property). We prove that $E$ is $p$-incompressible. Moreover, all smooth compactifications of $E$ (including those given by toric varieties) are $p$-incompressible. The main requisites of the proof are: (1) A. Merkurjev's degree formula (requiring the characteristic assumption), generalizing M. Rost's degree formula, and (2) combinatorial construction of a smooth projective fan invariant under an action of a finite group on the ambient lattice due to J.-L. Colliot-Thélène–D. Harari–A. N. Skorobogatov, produced by refinement of J.-L. Brylinski's method with a help of an idea of K. Künnemann.