Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?

Edit: Further cases of "fractional motives" as discussed in the article above (but with other weights) are expected to arise in quantum cohomology, say the experts. I wonder how the idea of such new motives may fit into the "usual" connections between motives, l-adic representations, periods and values of L-functions?