Generalization of geometric delineability without the relative-dimension-one condition

Let $R$ be the real closed field under consideration, let $X$ and $Y$ be affine $R$-varieties, and let $pi:X\to Y$ be a morphism such that $pi_*\mathcal{O}_X$ is a free $\mathcal{O}_Y$-module of finite rank. Suppose $S\subseteq Y(R)$ is a semi-algebraically connected semi-algebraic set on which $X_{\overline{y}}$ is invariant. Write $\pi_0^{-1}(S)$ for the relevant subset over $S$. Generalized geometric-delineability conjecture. There are semi-algebraic continuous functions $f_1,\ldots,f_s:S\to \pi_0^{-1}(S)$ satisfying

$$\pi\circ f_i=\operatorname{id}_S$$

such that $\pi_0^{-1}(S)$ is covered by the images of $f_1,\ldots,f_s$. This conjecture asserts that the geometric-delineability theorem remains valid without the Projection of Relative Dimension 1 condition.