疎遠, 不仲 and 絶交: distance, conflict and ending a friendship
Compare growing distant, an untrue rumor of conflict and a wish not to end a friendship. Keep relationship states, a negated rumor and a story's declaration distinct from one confirmed breakup.
An old friend becoming distant and a meal with a brother
「引っ越してから昔の友人と疎遠になった。」 (After moving, I became distant from an old friend.)
Another example reports having a meal with an estranged older brother for the first time in years. 疎遠 describes distance in the relationship; it does not itself report a friendship formally ended or a family relationship erased.
A misunderstanding and an untrue rumor
「兄弟は長年の誤解で不仲になった。」 (A long misunderstanding damaged the brothers' relationship.)
A different sentence says the rumor that two people did not get along was untrue. A relationship damaged by misunderstanding and a negated rumor are separate scenes.
Not wanting to end a friendship and a fictional declaration
「小さな誤解だけで友人と絶交したくない。」 (I do not want to end a friendship over a small misunderstanding.)
In a story, two people declare their friendship over after an argument. Keep that declaration separate from the speaker's wish not to end a friendship over a small misunderstanding.
Does 二人が不仲だという噂は事実ではなかった confirm that the two people did not get along?
Show the answer and why
No. It says that rumor was untrue. Preserve the negation rather than turning the rumor into confirmation of conflict or a completed breakup.