A hardware device taking an integer SDI/HDMI input can by definition only support a subset of CLF. For instance, any LMT designed to operate on linear float ACES2065-1 data cannot be used directly in a LUT box, as linear float data cannot be sent over SDI. It would be necessary to log encode the SDI data, and the LMT would need to be modified so that it had an appropriate conversion from log added at the start, and back at the end.
But is this likely ever to happen?
The normal situation in a LUT box is more likely to be that the LMT is concatenated with an Input Transform for e.g. LogC/AWG, and an Output Transform for e.g. Rec. 709. Then it can produce a video signal for on set monitoring which embodies the look of that LMT. But I would say it is not true to say the new LUT is the LMT. It is a new transform which applies the LMT.
So I would say that CLF can store a LUT formatted for the capabilities of a particular LUT box, e.g. 1D + 3D + 1D. But no LUT box can “implement CLF” fully. Nor does it need to.