Fix integers $1 \leq k \leq n$ and suppose $\mathbf{x} \in \mathbb{R}^n$ is such that $e_j(x_1,x_2,\ldots,x_n) \geq 0$ for all $1 \leq j \leq k$, where $e_j$ is the $j$-th elementary symmetric polynomial.

Question 1:Is it true that $x_1 + x_2 + \cdots + x_{n-k+1} \geq 0$?

The answer is trivially "yes" in the edge cases $k = 1$ and $k = n$, but the intermediate cases seem much less obvious.

While Question 1 is the one that I'm really interested in, there is a natural generalization of it that is perhaps known, so I'll ask it now:

Question 2:Is it true that $e_j(x_1,x_2,\ldots,x_{n-1}) \geq 0$ for all $1 \leq j \leq k-1$?

In other words, can we use non-negativity of elementary symmetric polynomials in $n$ variables to infer non-negativity of elementary symmetric polynomials in $n-1$ variables? If the answer to Question 2 is "yes" then we can use it $k-1$ times to see that the answer to Question 1 is "yes" too.