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We show that for every $0<\delta <1$, the Fourier extension conjecture on the paraboloid in three dimensions is equivalent to a local single scale smooth Alpert trilinear inequality of the form:

$$\begin{align*}&\left\Vert \mathcal{E}\mathsf{Q}_{s,U_{1}}^{\eta}f_{1}\ \mathcal{E} \mathsf{Q}_{s,U_{2}}^{\eta}f_{2}\ \mathcal{E}\mathsf{Q}_{s,U_{3}}^{\eta} f_{3}\right\Vert _{L^{\frac{q}{3}}\left( B_{\mathbb{R}^{3}}\left( 0,2^{\frac{s}{1-\delta}}\right) \right) }\\ &\quad \leq C_{\delta,\varepsilon,\nu}2^{\varepsilon s}\left\Vert f_{1}\right\Vert _{L^{\infty}\left( U_{1}\right) }\left\Vert f_{2} \right\Vert _{L^{\infty}\left( U_{2}\right) }\left\Vert f_{3}\right\Vert _{L^{\infty}\left( U_{3}\right) }, \end{align*}$$

which is a variant of the analogous multiscale trilinear inequality in Rios and Sawyer (2025, Equivalence of linear and trilinear Kakeya conjectures in three dimensions), where the smooth Alpert projections $\mathsf {Q}_{s,U_{k}}^{\eta }$