the Sheitel Or the Shofar, One Or the Other

A community in Jerusalem - a community of extremists, it seems - has put up a notice in its shul, Ohel Yesharim, for the High Holidays letting people know that they will not be facilitating women in fulfilling their obligations if they cover their head with a wig rather than with a "head covering".
sources: Haredim10, Kikar
1. I presume "head covering" means tichel or snood or something like that. Technically the head is covered equally when in a wig, even if you believe a wig is immodest.
2. what obligations do women need to fulfill that they are preventing? The articles (linked above) assume it refers to shofar blowing, but that is not mentioned explicitly in the sign, and, more importantly, women are not obligated to hear the shofar. Unfortunately I cannot think of another example of what it might refer to.
3. My approach would be different. In at least 2 ways:
        1. I would not consider a person a sinner and beyond the pale for doing something that I thought was wrong, if they had other valid opinions to rely on. I might not agree, but I wouldn't consider them sinners.
         2. Especially on the High Holidays, when we are asking God to ignore our sins or forgive them and continue to include us within His community of the Jewish people, my approach would to try to be more inclusive than exclusive.
4. I guess they have moved on from the smartphones as the focus of these types of exclusions.
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