Step 1: Sentence Comprehension. The sentence conveys a scenario of two individuals experiencing a dispute or estrangement stemming from a trivial matter. The appropriate phrase should signify a division or separation occasioned by this issue.
Step 2: Option Analysis.
- "Down" is unsuitable as it pertains to direction or intensity, not conflict.
- "Out" is the correct selection because "fall out" is a standard idiom denoting a disagreement or severed relationship.
- "Into" signifies entry, which is irrelevant to the context.
- "By" relates to proximity or method, neither of which applies here.
Step 3: Resolution.
"Fall out" accurately describes a disagreement between individuals.
Therefore, the correct option is 2. out.
Final Answer:
\[
\boxed{\text{The correct answer is 2. out.}}\]