I see this claim repeated a lot but you didn't say where it comes from so it's a bit hard to discuss. In particular, you are making a claim about the productivity of low earners but I don't think this is something that is measured in the US? My understanding is the BLS computes industry-wide productivity measures, ie mean productivity (as opposed to say median productivity), and one would expect this distribution to be significantly right-tailed and the mean to be mostly influenced by the right tail.
I haven't seen any data that breaks down productivity growth by income, so if you have a source for your claim I'd love to update myself. When I search for "productivity growth by income" I see versions of your claim but again they are all about population _means_.