It’s not differing by a constant, at least in some situations.
Here’s interstice’s comment below, reproduced:
I only just realized that you’re mainly thinking of the complexity of semimeasures on infinite sequences, not the complexity of finite strings. I guess that should have been obvious from the OP; the results I’ve been citing are about finite strings. My bad! For semimeasures, this paper proves that there actually is a non-constant gap between the log-total-probability and description complexity. Instead the gap is bounded by the Kolmogorov complexity of the length of the sequences. This is discussed in section 4.5.4 of Li&Vitanyi.
It’s not differing by a constant, at least in some situations.
Here’s interstice’s comment below, reproduced:
Link to the paper is below:
https://www.cs.bu.edu/faculty/gacs/papers/Gacs81.pdf