R' Brody discusses the issue on JPost's Ask the Rabbi. He cites many authorities, such as R.Y.D Soloveitchik and R.Y.Henkin, that permitted it and R.M. Feinstein who said it was common practice in Europe. However, he mentions some objections raised:
The permissive position was opposed by many decisors, who argued that
(a) women could not participate in this part of the service (Minhat
Yitzchak 4:30), (b) the classic mystical sources only spoke of men’s
recitation (Mishpatei Uziel OC 3:13), (c) it was immodest in public
settings (Aseh Lecha Rav 5:33), or (d) it would ultimately support
antinomian trends found in the non- Orthodox movements (Yahel Yisrael
2:90).
It may be that in certain areas the Reform and Conservative movements cause an opposite reaction from Orthodox Jews.