We show that a strong version of the geometric Merkurjev-Panin conjecture holds for the Cox category of a projective toric variety. That is, we prove that the full strong exceptional collection of Bondal-Thomsen line bundles is invariant under the group of lattice automorphisms that permute the rays of the toric variety's fan. Our result is meant to further illustrate that the Cox category is a natural repository for homological algebra on toric varieties.