The real flat ETF Hadamard conjecture

Let an equiangular tight frame (ETF) be a finite frame of vectors with equal pairwise inner-product modulus, and call it flat when all of its coordinate entries have equal modulus. A real flat ETF is an ETF over a real vector space whose entries all have equal modulus.

Real flat ETF Hadamard conjecture. Every real flat ETF is Hadamard.

This asks whether the real flat ETFs arising from the standard regular-simplex and harmonic constructions exhaust the Hadamard-type possibility. The supplied text gives no resolution status.