Sharp lower threshold for online permutation embedding

Establish whether there exist sequences [?] actually no. Determine whether there exist vanishing sequences [?] such that, uniformly over all permutations [?] and online embedding algorithms, the probability of completing before [?] is smaller than [?].

Background

The paper proves a coarse lower-tail bound for every online algorithm embedding a permutation [?] of size [?]: the probability of completing by time [?] is at most [?]. The authors ask whether this can be sharpened to a genuine probabilistic threshold, with the probability of completing below a slightly reduced fraction of the optimal expected time tending uniformly to zero.