Naive bilinear Kakeya conjecture for transversal Heisenberg tubes

Let $c\geq 0$ be a small constant, let $e_1,e_2$ be the standard basis vectors in $\mathbb{R}^3$ parallel to the $x$- and $y$-axes, and let $\delta\in(0,1)$. A pair of families of Heisenberg $\delta$-tubes $(\mathcal{T}_1,\mathcal{T}_2)$ is transversal if, for every $T_j\in\mathcal{T}_j$, $j=1,2$, the horizontal direction $e(T_j)$ of $T_j$ satisfies $|e(T_j)-e_j|\leq c$. Let $|\mathcal{T}_j|$ denote the cardinality of $\mathcal{T}_j$, and let $\chi_{E_j}$ be the indicator of the union of the tubes in $\mathcal{T}_j$.

Naive bilinear Kakeya conjecture. Let $\varepsilon>0$. For every $\delta\in(0,1)$ and every pair of transversal families of Heisenberg $\delta$-tubes $(\mathcal{T}_1,\mathcal{T}_2)$, the corresponding bilinear Kakeya estimate holds with the natural exponent $3/4$, up to a factor of $\delta^{-\varepsilon}$.

This is the bilinear analogue of the Heisenberg Kakeya inequality, motivated by the multilinear estimates of Bennett, Carbery, and Tao. The supplied candidate span omits the displayed inequality itself, so the precise measure, normalization, and dependence on the tube families cannot be recovered from the available text.