Background
It can be challenging to find computational problems that are solvable in $DTIME(n^k) - DTIME(n^{k-1})$ where $k \geq 2$.
Although some natural problems are known to exist, many of them are in someway equivalent to simulating Turing machines or the lower bound result is heavily dependent on the machine model.
See this question for more details: Problem in deterministic time $n^p$ and not lower
Question
Does there exist $k \geq 2$ such that there exist problems in $DTIME(n^k) - DTIME(n^{k-1})$ that are not computationally hard for $DTIME(n^{k-1})$ under nearly linear time reductions?
Update: As pointed out by @NealYoung in the comments, when $k=2$, no such problems exist. Also, for the computational model, let's go with multitape Turing machine.
Why does it matter?
I suspect that if the answer is false (meaning that no such problems exist), then all polynomial time problems have non-uniform nearly linear size circuits.
Let me try to explain.
Let $k \geq 2$ be given.
Suppose for a minute that every problem in $DTIME(n^k) - DTIME(n^{k-1})$ is hard for $DTIME(n^{k-1})$ under nearly linear time reductions.
Next, consider a problem $X$ that is complete for $DTIME(2^{k \cdot n})$ (such as the problem of simulating a $2^{k \cdot n}$-time bounded Turing machine on an input). By the time hierarchy theorem, this problem cannot be solved in much less than $2^{k \cdot n}$ time.
Then, convert this problem $X$ from binary inputs to unary inputs to get a problem $X'$. We have that $X' \in DTIME(n^k) - DTIME(n^{k-1})$. By the assumption, it follows that $X'$ is hard for $DTIME(n^{k-1})$ under nearly linear time reductions.
Finally, we can build small non-uniform circuits for $X'$ because there are so few possible unary input strings. Also, we can build a small circuit for any nearly linear time reduction. Combining these together, we get small circuits for all problems in $DTIME(n^{k-1})$.
Maybe we can get a contradiction based on some known circuit lower bounds? Or, there could even be relativized results that come into play. Any thoughts are greatly appreciated. Thank you!