Identifier equality of identifiers stashed in preserved syntax properties

In the language I am working on, I store identifiers in preserved
syntax properties to attach type information to pieces of syntax.
More specifically, I store identifiers inside prefab structs, which
are then stored within preserved syntax properties. This seems to
work fine when my program spans a single module, but it breaks in
a confusing way when I use an identifier from another module.

Specifically, I have two identifiers from the “same” place, but the
macro that attaches the syntax property is used in two different
modules. I would expect these identifiers to be free-identifier=?,
but they are not. Looking at what identifier-binding returns, I get
the following results:

  (#<module-path-index:()> List #<module-path-index:()> List 0 0 0)
  (#<module-path-index:(rascal/prelude)> List #<module-path-index:(rascal/prelude)> List 0 0 0)

Obviously, the module path index values are different, but I don’t
know why or how to fix it. My attempt to create a minimal reproducible
test case led to me filing racket/racket#1495[1], but that issue
seems to be slightly different from my problem.

If I stick an identifier in a preserved syntax property, what are
the guarantees I have about it? Is using free-identifier=? on it
from another module dangerous in some way? The documentation doesn’t
seem to mention anything about this, but in the meantime, I’m
honestly a little unsure how to even begin to fix this (aside from
radically changing how my program works).

[1]: https://github.com/racket/racket/issues/1495

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