Moderated Mediation when Independent Variable is also the moderator

I have a question on a special case of moderated mediation...
I want to compare some effect sizes of: a "simple moderation model" with a "moderated mediation model in which the IV is also a moderator".
I don't have data yet. But I want to know if the betas (effect sizes) explained in the attached pdf are different from each other or not?

in fact the second model (only moderation) exists in the literature that I am studying.
Can I argue that the model is misspecified because they have not included X in the model? and Can I claim that if we follow moderated mediation model the effect of C on

If I consider the "moderated mediation model" with two regressions, the second regression is exactly the same as the the regression for the "moderation model".

And if we analyze them obe-by-one, beta4 and beta2 in two models will be the same. What if we do SEM?

Summary of my Question:
are beta2 and beta4 in model 1 and model 2 remain the same ??

Thank you!


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