Conjectured sharp exponent 2/3 for the bilinear Heisenberg Kakeya inequality

Prove that the sharp L^p exponent in the bilinear Heisenberg Kakeya inequality of Theorem 1.1 equals 2/3; that is, show that the inequality holds with exponent p=2/3 and cannot hold with any smaller exponent.

Background

Motivated by combinatorial and incidence-geometric heuristics (including parallels with Szemerédi–Trotter and point–line duality), the authors conjecture that the optimal L^p exponent in Theorem 1.1 is 2/3.

They provide lower bounds showing that any valid exponent must be at least 2/3, and thus the conjecture pinpoints the expected sharp threshold.