I believe this is true:

Suppose $G$ is a graph. If $G$ has a subdivision of a large binary tree, prove that $G$ has an induced subgraph which is a subdivision of a large binary tree or the line graph of a subdivision of a large binary tree.

However, I think it is very difficult to prove in general, so I was hoping to do it for the case when $G$ has bounded treewidth (or say really small treewidth). Any ideas for those cases? Thank you!

[EDIT: I initially forgot the word subdvision in the question]

[EDIT2: PROVED, someone can delete]