Generalized coarse Bézout conjecture for persistent zero sets

Let $M$ be the compact manifold, let $D$ be the relevant geometric parameter, and let $\mathcal{F}_{\lambda_j}$ denote the corresponding eigenspaces. Let $f_j\in\mathcal{F}_{\lambda_j}$ and let $s=(f_1,\dots,f_n)$ be the section considered in the coarse Bézout theorem. For $\delta>0$, write $z_0(s,\delta)$ for its persistent zero count, and let $k>n/2$ be the integer used there. Generalized coarse Bézout conjecture. One expects

$$z_0(s,\delta)\leq \frac{C_1}{\delta^{n/k}}\left((\lambda_1+1)\cdot\ldots\cdot(\lambda_n+1)\right)^{\frac{1}{q}}+1,$$

where $C_1$ depends only on $M,D,k$. This would extend the coarse Bézout estimate from comparable eigenvalues to functions with distinct eigenvalues, reflecting the product of the corresponding algebraic degrees. The source presents this as a tempting conjecture rather than an established result; the status of the general estimate is therefore open.