If $f \neq 0$, do we have $\int_{ x_i \geq 0} f(x) e^{-\sum_{i=1}^n c_i x_i^2} d^nx \neq 0$ for some $c \in (0,\infty)^n$?

Question

Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a bounded measurable mapping. Next, define \begin{equation} \mathbb{R}^n_+ := \{ x \in \mathbb{R}^n \mid x_i \geq 0 \text{ for all } i=1,\cdots,n\} \end{equation}

Now, assume further that $f(x) \neq 0$ for a subset of $\mathbb{R}^n_+$ of positive measure.

My question is now that:

is it true that \begin{equation} \int_{ \mathbb{R}^n_+} f(x) e^{-\sum_{i=1}^n c_i x_i^2} d^nx \neq 0 \end{equation} as an element of $\mathbb{R}^m$ for some $c=(c_i) \in (0,\infty)^n$?

I think this is essentially an extension of my previous question If $f(x) \neq 0$ on $[0,\infty)$, does there exists some $\alpha \in [0,\infty)$ such that $\int_0^\infty f(x)e^{-\alpha x^2}dx \neq 0$? but seems trickier than expected..

Could anyone help me?