[math-fun] topswops

Hi folks, I'm amateurishly working on the topswops problem (as in A000375 and A000376 of OEIS). Does anyone know anything about the state of the art in lower and upper bounds for terms beyond those given in the OEIS? I'd like to poke around and improve some of the lower bounds if I can, by actually constructing permutations. Thanks, --Joshua Zucker