The expected contribution from isolated zeros of the vector field

Let $f_l$ be the random spherical harmonic and let $V$ be the vector field on $S^2$ considered above. Writing $DF_l$ for the derivative of the map $F_l=(f_l,Vf_l)$ and $\det^{\perp}DF_l$ for its normal Jacobian, the expected integral over the simultaneous zero set $\{f_l=0,Vf_l=0\}$ is the quantity displayed above. Expected-value conjecture. The expected value above should satisfy

$$\mathbb{E} \left[ \int_{ \{f_l=0, Vf_l = 0 \}} | \det^{\perp} DF_l(x) | \, d\sigma(x) \right]=\mathcal{O}(l^{-1}).$$

This estimate would show that the contribution from the complement of the region $U$ is negligible compared with the previously relevant scale $\mathcal{O}(l^{3/2})$. The parser supplies no evidence that the estimate has been proved or disproved.