I proved both $latex (\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}$ and $latex (\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}$.

So there is an error in my math research book.

I will post the details of the resolution as soon as I will locate and correct the error. While the error is not yet corrected I have added a red font note in my book.

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