Why does temperature scaling preserve top-1 predictions?

For multiclass logits z and a learned scalar temperature T>0, temperature scaling computes softmax(z/T). Why does this normally preserve the predicted class?

The learned temperature is constrained to equal one whenever the classifier is accurate

Dividing every logit by the same positive scalar preserves their ordering and therefore their argmax

Softmax probabilities are invariant to every positive scaling of their logits

Temperature scaling changes only the bias term of the final layer