Deteremine if Language is in $R$ or $RE$

$$L =\left \{ \langle M \rangle \mid \exists x\in \Sigma^* \left(\left | x \right |\leq 10000 \wedge H(M, x\right) \right \}$$

Where $$H(M, x)$$ denotes whether Turing machine $$M$$ halts on input $$x$$.

My guess is neither $$R$$ or $$Re$$ because I would have to run infinite TM's in order to find out if every TM halts on x or is not halting on x. Is my intuition correct?

The language is in RE: simply generate all inputs $$x$$ with $$|x|\le 10000$$ (they are finitely many), and simulate $$T$$ on all inputs in parallel. Whenever one simulation halts, accept.
The language is not in co-RE, as otherwise it would be decidable and we could solve the Halting problem. Indeed, to decide whether a Turing machine $$T$$ halts on empty input, we could create a new machine $$M$$ that simulates $$T$$ if $$x=\varepsilon$$ and does not halt otherwise. Then $$M \in L$$ if an only if $$T(\varepsilon)$$ halts.