How to pronounce "-ers" ending words (r-colored vowel)

I searched through some phonology books and the Internet more than two hours to find out the answer about "-ers" ending pronunciation. As I know, the suffix "-s" is pronounced as "/z/" if the last sound is a voiced sound such as dogs, cans, and beans. However, I can't find the answer about how to pronounce "-s" after r-colored vowel words like hackers, fathers and farmers. I guess it would be pronounced as /z/ since every vowel is voiced and the retroflex "r" is also voiced, so the r-colored vowel "ɚ" is also voiced one. Nevertheless, I have found many results that "-ers" ending words I mentioned above (hackers, fathers and farmers) are pronounced as "/ɚs/", not "/ɚz/." This makes me really confused. Please let me know if I know correctly or not.

The plural -s is always pronounced /z/ after [ɹ] and [ɚ] can be analysed as [əɹ] so the plural -s should also be pronounced /z/ after [ɚ] and [ɝ]

However, there are some words in which the s is pronounced /s/ such as purse, curse etc., in which case, the s is pronounced /s/ after the rhotacised vowel.