Existence of cyclic 2-designs in every dimension at least 3

Let a cyclic $t$-design in dimension $d$ be a finite orbit of a unitary acting on quantum states whose decohered probability vectors form a simplex $t$-design. Cyclic 2-design existence conjecture. For $d\geq 3$, a cyclic $t$-design exists for $t=1$ or $t=2$. The claim is presented as a corollary of the preceding conjectural lower-bound and monotonicity discussion; cyclic 2-designs are constructively known in the dimensions covered by the explicit construction, but the assertion for every $d\geq 3$ remains open.