Sharp exponent conjecture for the bilinear Heisenberg tube estimate

Let the bilinear tube estimate in Theorem $1$ be understood with its stated $L^p$ exponent, and let the sharp exponent mean the smallest exponent for which that estimate holds. Sharp exponent conjecture. The sharp $L^p$ exponent in Theorem $1$ is $2/3$. The paper states that the theorem currently has exponent $3/4$, while the preceding proposition gives only the lower bound $p\geq 2/3$. Thus the conjecture predicts that this lower bound is optimal and that the exponent $3/4$ can be improved.