Let $\mathbb B^n$ be a unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\mathbb B^k$ be a unit ball in $\mathbb R^k$.

Question. Is it true that $F$ has a critical point in the interior of $\mathbb B^n$ if there exists a smooth surjective map $\varphi:\mathbb B^n\to \mathbb B^k$, that has the following properties:

1. For any point $x$ in the interior of $\mathbb B^k$ the minimum of $F|_{\varphi^{-1}(x)}$ is attained in the interior of $\mathbb B^n$.

2. The maximum $$\max_{y\in \mathbb B^k}\min_{x\in \varphi^{-1}(y)} F(x)$$ is attained in the interior of $\mathbb B^k$ at some point $\bf y$.

3. Moreover, the strict minimum of $F$ on $\varphi^{-1}(\bf y)$ lies in the interior of $\mathbb B^n$.

Comment. I hope that the answer is positive if the differential of $\varphi$ is surjective on the whole ball $\mathbb B^n$ (but the comments of Fedja were producing counter-examples to the previous versions of the question).

I would be grateful for a reference (or a counter-example...).