Concurrent TS/SCI Investigations?

So as I have stated on this forum previously, I’m currently in the process for a TS/SCI FSP position and have been since January.

I recently found out that another intelligence agency may be giving me a CJO, which will require the same TS/SCI FSP clearance. Can they run their own investigation concurrently?

I mentioned the existing and ongoing investigation to this latest agency, but they didn’t indicate what would exactly happen. Would this latest agency be able to use some or all of the already in-process information?