Sharp L^p exponent for the bilinear Heisenberg Kakeya inequality

Determine the minimal exponent p for which the bilinear Heisenberg Kakeya inequality of Theorem 1.1 holds; specifically, ascertain whether the exponent 3/4 in the bound ∫_{R^3} (∑_{T_1∈𝒯_1} χ_{T_1})^{3/4} (∑_{T_2∈𝒯_2} χ_{T_2})^{3/4} ≲_ε δ^{4−ε} ( #𝒯_1^{3/4} #𝒯_2^{3/4} + #𝒯_1 + #𝒯_2 ) (under the stated transversality and (δ,α)-broadness hypotheses for the tube families) is sharp, or whether it can be improved to a smaller exponent p.

Background

Theorem 1.1 establishes a bilinear Kakeya inequality for Heisenberg δ-tubes with a left-hand side featuring the exponent 3/4. The authors provide lower-bound examples showing that any admissible exponent must be at least 2/3, but they do not know if 3/4 is optimal.

They relate this question to incidence geometry bounds used in the proof, and later formulate a conjecture that the sharp exponent should be 2/3, motivated in part by Szemerédi–Trotter-type heuristics and point–line duality in the Heisenberg setting.