Existence of d by 2d complex equiangular tight frames in every dimension

Prove that for every positive integer d ≥ 2, there exists a complex equiangular tight frame consisting of 2d vectors in complex dimension d.

Background

The paper focuses on ETFs with twice as many vectors as the ambient dimension, a parameter range connected to Hadamard and conference matrices and also accessible through doubling, multiplication, and power constructions. Fallon and Iverson’s conjecture asserts existence in every dimension d ≥ 2. The paper verifies additional infinite families and specific dimensions but does not resolve the conjecture in full.