Sparse feasible solution $|x|_0\le k$ for system of linear inequalities $A x \le b$

Suppose the set of linear inequalities $$Ax\le b$$, in which $$A\in\mathbb{R}^{m\times n},x,b\in\mathbb{R}^n$$ is given. Is it possible to determine in polynomial time with regard to $$m$$ and $$n$$ if there is a feasible solution that is $$k$$-sparse, i.e. there are only $$k$$-nonzero elements in $$x$$?

What about if we know an upper bound on the $$L_1$$-norm of the solution $$|x|_1\le B$$. Would it be possible to answer the system has a sparse solution in polynomial time?

P.S. I'm aware of this question, but the answer does not address whether this approach always finds a feasible solution in polynomial time.