Sequential construction of all signature preorders without restarting

Establish whether the inductive construction yielding structured signature automata can be carried out sequentially to obtain all preorders without restarting the construction at each step (i.e., whether the iterative step always yields an x-structured signature automaton so the process can continue).

Background

The authors’ main constructive proof defines preorders iteratively and sometimes restarts from a smaller automaton to restore needed properties. They conjecture that restarting is unnecessary if one could ensure each step yields a suitably structured automaton, simplifying the construction.