The Clifford bound conjecture for unitary 2-designs

Let $d$ be the dimension of the underlying Hilbert space, and let a unitary 2-design be a set of unitary matrices whose second moments reproduce those of the Haar measure. The Clifford bound is

$$d^4-d^2.$$

Clifford bound conjecture. The cardinality of any unitary 2-design is at least

$$d^4-d^2.$$

The bound is attained by Clifford 2-designs in several dimensions and is known to be the smallest possible cardinality for designs based on the Clifford group. The conjecture asserts that no unitary 2-design, including one not based on the Clifford group, can be smaller.