Are there any context-free languages that are not known to be in $\mathrm{DTIME}(O(n))$?
Theorem 1 in "If the Current Clique Algorithms are Optimal,
so is Valiant’s Parser" shows an explicit context-free grammar $G$ such that if $G$ can be parsed in linear time (or even a bit slower), then there's a surprisingly fast algorithm for the $k$-clique problem. As the paper says, no such algorithm for $k$-clique is known to exist and it would be a breakthrough if it was discovered; therefore, there's no known linear-time algorithm for this grammar $G$.