(Not a member but native Persian speaker)
I think it used to be a bigger movement for it in early 2000 because there wasn't much support for Persian keyboard and script in software and lots of people had to write in English script to be able to use their phone but I haven't heard even the craziest people push for this idea recently because the support is fully there.
- There's no official place to "regulate" the language. Persian has APLL  that regulates both language and the script which we heavily use in fawiki (Our MoS is basically copy of the rules on writing )
- Unlike Tajik, the language is not native language of any group (or anyone for that matter).
- Tajik is not just a different script, it's quite different from "High Persian" (I don't know the correct translation of the term, I use Hoch Deutsch). For example, it has a significant number of loan words from Russian. I once tried speaking with a Tajik person in latin script, we gave up and switched to English.