  A rich line of work has been addressing the computational complexity of
locally checkable labelings (LCLs), illustrating the landscape of possible
complexities. In this paper, we study the landscape of LCL complexities under
bandwidth restrictions. Our main results are twofold. First, we show that on
trees, the CONGEST complexity of an LCL problem is asymptotically equal to its
complexity in the LOCAL model. An analog statement for general (non-LCL)
problems is known to be false. Second, we show that for general graphs this
equivalence does not hold, by providing an LCL problem for which we show that
it can be solved in $O(\log n)$ rounds in the LOCAL model, but requires
$\tilde{\Omega}(n^{1/2})$ rounds in the CONGEST model.
