A bit more details: I'm trying to prove that if the first condition holds, then the following statement is also true: $$\exists z\neq 0 \text{ such that }z\in K°\cap L^\perp.$$
That second condition allows me to use other results I have on the context of my problem and get some really nice results. Notice that the polar of a linear subspace is its orthogonal complement. My intuition tells me it's true, and is easy to see that in $d=2$ that is the case. However, I have not been able to prove it. I believe some variation of Farka's Lemma might be useful. I also tried to use the fact that the intersection of polar of cones is the polar of their Minkowski sum ($K_1^°\cap K_2^°=(K_1\oplus K_2)^°$), also to no success.
Any help is welcomed and appreciated.
Edit: As Theo commented, there is an easy counterexample. I should add the restriction that $K$ is not a subspace.