CSS not applied to all FSE templates when Blocks CSS plugin is used

Description

If i use the Blocks CSS plugin and set a background color in the header template part, it will be applied only on the homepage. If i deactivate the plugin and switch to Otter, the CSS will be applied to all templates (homepage, page, single post...). Here is a video with the behaviour https://vertis.d.pr/v/ldxpHX

Reported here - https://wordpress.org/support/topic/css-for-template-work-on-home-page-but-not-work-on-other-pages/

Step-by-step reproduction instructions

1. Install Blocks CSS plugin and Raft
2. Add some Custom CSS in the header template part
3. Check if it's applied on Page, Single post
4. Deactivate Blocks CSS and activate Otter
5. Check if the Custom CSS is applied on Page, Single post

Screenshots, screen recording, code snippet or Help Scout ticket

https://vertis.d.pr/v/ldxpHX

Environment info

WordPress 6.4.1 / PHP 8.0 / Blocks CSS 2.5.2 / Otter 2.5.2

Is the issue you are reporting a regression

No

Start by reproducing the issue with WordPress 6.4.1, PHP 8.0, Blocks CSS 2.5.2, and Otter 2.5.2 using the header template part and the linked recording. Compare CSS output on the homepage, pages, and single posts with Blocks CSS enabled and with Otter enabled; done means the custom CSS applies consistently across those templates.