Location of the first failing exponent

Determine whether p₀=4, where p₀=inf{p>2:e(ℓₚⁿ)>n+1 for some n}, equivalently determine whether Kusner’s conjecture begins to fail immediately above the range 2≤p≤4.

Background

The paper proves that the n+1 bound holds for every 2≤p≤4, while its certified construction at p=5 and its continuation argument imply that failure occurs for some exponents below 5. Consequently, the threshold p₀ at which a violation first appears lies in [4,5).

The unresolved question is whether the lower endpoint is sharp: if p₀=4, then violations would occur arbitrarily close to 4 from above, despite the established validity throughout the closed interval [2,4].