first of all, the work your company does is awesome!
By browsing the showcases I went in love with the UI the animations and the infographics.
If I well understood the topic, the conversion rate is the sum of the valid contact forms you receive, right?
I think that the floating CTA for activating the contact form could be a good starting point.
I would reuse the room you gave to the “settings” icon.
I’m not 100% sure this add value to the whole experience, maybe you tested it and I would love to hear what’s the reason to have it there.
On top of that I would add a simple section in any project detail pages to invite the users to provide their emails in order to receive more details on that specific project/topic.
As a dumb scenario, it could be that I’m a Product Owner of a company that operates in the cyber security field and I don’t care at all of the weather forecast.
I would think on a couple of options, create a quick and dirty wireframe (I love your low fidelity design too) and then I would validate both of them with a qualitative usability test.
Good luck with it!