The String itself is definitely throwing a wrench into the progression of their relationship, but I think it's also important, because of the fact it's throwing a wrench into the progression of their relationship.
The trick will be whether Tsumugi can let her attraction to her senpai exist independent of and despite the String coming into play, and whether her senpai can square that she loves Tsumugi for her own sake, with the String being only incidental and not the root cause of her falling in love.
Basically--rather than the String being the impetus to their love, it's affirming it. But that perspective hasn't been sorted through yet, as it seems the general sentiment and perception of everyone around them is that the String dictates who falls in love. The String appears before any relationship happens, and it only happens because the String appears.
So the issue I see, is that Tsumugi assumes the String invalidates the concept of organic love, when it could simply complement it. And part of that is the societal conditioning of all the stories and incidents everyone can see.
But its appearance between the two of them, doesn't invalidate any love they'd otherwise feel, and it will be a matter of decoupling the String's meaning from the feelings they have, and trusting that their relationship is built on its own merit of who they are to the other.
Which takes time, of course. It's effectively rewiring one's brain and way of thinking. Plus, it's only been 4 chapters, so I don't expect it to happen immediately or cleanly.