Order-Preserving Bijection $f:A\to A^*$?

Question

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ordering make $A^*$ well-quasi-ordered as well (reference to Corollary $1.7$ of [2]).

References:

1. http://www.math.harvard.edu/~lurie/155notes/lecture19.pdf

2. https://research-repository.st-andrews.ac.uk/bitstream/10023/7963/1/BCCpaperv9.pdf

3. http://arxiv.org/pdf/1107.5070v2.pdf

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Crossposted on MathOverflow.

Discussion: https://mathoverflow.net/a/230193/82691