If two distributions have the same moments, how different can they be? Let us suppose we have two distribution functions $F$ and $G$ with shared domain and also shared moments but not necessarily shared moment-generating functions.
I have seen from "Whether distributions with the same moments are identical" that two distributions can have the same moments while being distinct. I also saw from this answer the following plot.

While the distributions above are clearly different, even with identical moments, there is an overall 'shape' that each of these distributions roughly follow. This led me to the hypothesis that distributions with identical moments cannot be boundlessly different. Perhaps that is true, or perhaps not.
Defined either by a metric or a divergence (answerer's choice), is there an upper bound on how different $F$ and $G$ can be from each other when their moments are the same? If so, what is it?

See Jordan Stoyanov's "index of dissimilarity" for one such metric, in his paper, ``Stieltjes Classes for Moment-Indeterminate Probability Distributions,'' Journal of Applied Probability, v. 41(A), pp. 281 - 294, 2004.
