Generalized analytic delineability conjecture for iterated resultants

Let $f_1,\ldots,f_t$ be relatively prime real polynomials of positive degrees with respect to $x_{f+1},\ldots,x_r$, and let $x=(x_1,\ldots,x_f)$. Let $D(x)$ be a generalized discriminant of $f_1,\ldots,f_t$ with respect to $x_{f+1},\ldots,x_r$, with $D(x)\neq 0$. Let $S$ be a simply connected submanifold of $\mathbb{R}^f$ such that the total number of common zeros, counted with multiplicity, of the $f_i$ in $\mathbb{C}^t$ is finite and constant on $S$, and $D(x)$ is order-invariant on $S$. Suppose further that the polynomials $f_i$ and their first derivatives with respect to the last $t$ variables have no common zero in the cylinder $S\times\mathbb{R}^t$ over $S$. Generalized analytic delineability conjecture. Then the variety $V$ of the $f_i$ is analytically delineable on $S$ with respect to the last $t$ variables. This conjecture would provide the first of two lifting theorems needed to extend CAD-based quantifier-elimination methods from the cases $p=t=1$ and $p=t=2$ to the general family $p=t\geq 1$; the required generalized discriminant and analytic-delineability notions are proposed by analogy with the previously studied cases, so the conjecture's resolution is not established here.