I added the integration and a test at the level of assemble. What are your thoughts?

I think it looks great. I was wondering if you'd consider integrating @tasks with a scheduler in the :dofsplitting variant? It might also be worth exploring using OhMyThreads chunks() for splitting the basis functions. This could help unify the API between :dofsplitting and :cellcoloring, making the interface more consistent across both approaches. That said, I completely understand if you'd prefer to stick with the current implementation

Thanks. I completely agree with fixing the dofsplitting code to allow for robust task migration. This was next on the agenda. I will need to do this as well for the quasi-local integral operators where a non-trivial reduction step is involved in order to produce the final matrix. I'll take the cue from how you dealt with task-local storage!

I want to merge but there is a conflict in runtests.jl

Not sure how to proceed using just rebase (as opposed to merge). What is the prescribed procedure? Copy the branch, rebase, and start a new PR?