Implement a function balance_undersample that takes a list of (sample, label) tuples and returns a new list where each class has the same number of samples, equal to the count of the minority (least frequent) class.

Rules:

• Determine the minimum count across all class labels.
• For each class, keep only the first min_count samples in their original order of appearance.
• The returned list should preserve the original ordering: iterate through the input once and include a sample if its class has not yet reached min_count kept samples.
• If the input is empty, return an empty list.

This mirrors a common preprocessing step before training a classifier on an imbalanced dataset.