What's the complexity of `+=` for a string in Python

Question

Since it's immutable it should be O(n) however some optimisations are done under the hood (cf. other SO posts like this one https://stackoverflow.com/a/70499811/11092636).

However, since it's neither O(1) neither O(n) (source, the benchmark I just did below), what is the complexity? Are there official sources?

MRE to reproduce:

import time
import matplotlib.pyplot as plt

def average_time_to_append(length):
    total_time = 0
    for _ in range(20000):  # Repeat 20000 times
        test_string = 'a' * length
        start_time = time.time()
        test_string += 'a'
        end_time = time.time()
        total_time += end_time - start_time
    return total_time / 20000  # Return the average time taken

lengths = list(range(10, 5*10**5))

# Measure
times = [average_time_to_append(length) for length in lengths]

# Plot
plt.figure(figsize=(10, 6))
plt.plot(lengths, times, marker='o')
plt.xlabel('String Length')
plt.ylabel('Time to append (seconds)')
plt.title('Benchmark: Time to Append \'a\' to a String')
plt.grid(True, which="both", ls="--")
plt.savefig("test.png")