The $\mathsf{P} \neq \mathsf{PSpace}$ conjecture means that
There is a language $L \in \mathsf{DSpace}(O(n^t))$ for some $t>0$ such that for all positive integers $k$, $L$ requires $\Omega(n^k)$ deterministic time to decide.
But I need a stronger assumption
There is a language in $\mathsf{DSpace}(O(n))$ that requires exponential ($2^{\Omega(n)}$) deterministic time to decide.
Obviously the Exponential Time Hypothesis (ETH) implies the statement above: 3SAT is in linear space. But I don't want to assume ETH.
Is there a weaker conjecture (something that is plausible even if ETH is false) that I can use?