Infinitely many dimensions with excess equilateral sets in ℓ₅

Determine whether e(ℓ₅ⁿ)>n+1 holds for infinitely many positive integers n, and, if so, whether the excess e(ℓ₅ⁿ)−n remains bounded.

Background

The paper constructs an equilateral set of 58 points in ℓ₅⁵⁶, thereby proving e(ℓ₅⁵⁶)>56+1 and providing the first finite-p example exceeding the Euclidean n+1 bound. The construction establishes only a single witnessing dimension, n=56. The authors explicitly ask whether analogous violations occur in infinitely many dimensions and, if they do, whether the amount by which the equilateral number exceeds n is uniformly bounded.

The question is connected to the currently weak general upper bounds at exponent 5: the paper notes that no upper bound substantially better than order n log n is available there, while even linear growth of e(ℓ₅ⁿ) remains unresolved.