Existence of a 36-point projective toric MUB design in dimension six

There exists a uniformly weighted 36-point projective toric $2$-design in $P(T^6)$ satisfying the complete-MUB overlap condition. In the dephased finite model, this means there is an injective family $X$ of 36 unit-modulus phase vectors with zeroth coordinate one, whose exact degree-$(2,2)$ moments equal the projective-torus Haar moments and for which every distinct pair has squared overlap either $0$ or $6$.

By the proved dimension-six specialization of Theorem 4.4, this existence assertion is equivalent to the mission goal asserting seven mutually unbiased orthonormal bases in $\mathbb C^6$. It is therefore an alternative formulation of the unresolved existence problem, not an existence or nonexistence result.