Suppose that $L$ is a language recognized by a linear-bounded automaton with the constraint that it can only change each of its input cells at most $t$ times each, where $t$ is some constant integer. Must $L$ belong to $P$, the class of languages decidable in polynomial time? Even more stringently, does there exist a deterministic decider for $L$ that runs in $O(n)$ time, where $n$ is the size of the input? Of course, if we can answer the second question in the affirmative, then we can answer the first question as well.
One approach I'm considering to answering the first question is looking at the length-increasing grammar associated with $L$, and devising an algorithm that checks if this grammar is capable of generating its input in polynomial time.
I'm not sure how to approach the second question. Of course, the tape head for the constrained LBA is writing at worst $O(n)$ times for any input, but the computation on the input still may involve transitions that don't write to the tape.