[Fire door frame in staircase]

The stairs run into the living room with no corridor so no where else for it go, but the council have said a fire door is needed on there. Would strips routered into the door work as they would sit tight into the stairs string 

 

[Fire door frame in staircase]

Hi,

I’ve been asked to replace an existing door and frame to a fire door, and frame. 
I’m just wondering would the frame be ok fitting into the same position onto the staircase like the picture. Or cut into the staircase so the frame sits down tight to the step.

thanks