Great question, my company actually managed to accomplish this, but not the way you're thinking.
The short version is that it's probably impossible within normal Robot Framework, but we set up a persistent Linux VM and installed Jenkins and Robot Framework on it. This was then connected to the GitHub repository where builds are uploaded and set to run a new test on every push.
However, if you want to get into the internals of your particular browser, I found a way to guarantee that Chrome is started by Robot Framework maximized, so it might be possible to do the same thing, but in reverse. Assuming none of your keywords rely on xy coordinates, it should be fine. Robot Framework works just fine when the browser it's controlling is minimized.
Here's the code for what I did to start with.
Open Chrome
[Arguments] ${HOMEPAGE}
${options} Evaluate sys.modules['selenium.webdriver'].ChromeOptions() sys
Call Method ${options} add_argument --start-maximized
Create WebDriver Chrome chrome_options=${options}
Go To ${HOMEPAGE}
Edit: After further discussion with senior developers, the answer is no. It is not possible as asked because Selenium needs an available element to interact with. It might not necessarily be visible to the eye, but it needs to be available to the program, and that is not the case when the browser window is minimized. When I changed my above code to "--start-minimized", it opened minimized and then promptly ran visibly.