In my experience, EASA will only accept an application from the firm that already holds the FAA STC.
... and most of the US firms cannot be bothered to play ball with EASA, for what they see as a tiny market.
This has been an issue for years.
It's a pure restrictive practice, set up to support EASA Part 21 companies and to enable them to generate EASA STCs and retain the intellectual property on those STCs.