The Log Squared Appears

It degrades to $O(m \log n \cdot \alpha(n))$. A range add pulls the second maximum back apart from the maximum at a rate bounded by the inverse Ackermann function.

It degrades to $O(m \log^2 n)$. A range add can pull the second maximum back apart from the maximum, re-creating potential that the amortized argument had already spent.

Nothing changes, it stays $O(m \log n)$. A range add commutes with chmin, so the potential the amortized argument had already spent is never re-created.

It degrades to $\Theta(m\sqrt{n})$. A range add destroys the potential argument entirely, leaving only the trivial bound that one operation touches at most $n$ leaves.