Party Fence Dispute - GA

So I am new to this HOA Board, I was elected president this year, oh lucky me. So we are having a disagreement with our ARC, who tend to feel like they probably have more power than they actually do. Anyway, along our common area where the community playground/pool/clubhouse is there is a greenspace. Next to this is the first house in our neighborhood, there is a fence which needs to be replaced between the two, it is falling apart and has not been addressed by previous boards/owners in all the years past. I believe this would be considered a party fence between the community and the homeowner, thereby the community pays for half the fence cost. Our ARC feels differently, they do not believe the community should be responsible for the cost because the fence sits 3ft onto their property and it would set a precedent. Thing is, there are no other homes in the neighborhood that borders a common space area, only perimeter fencing which is shared with the road. They say if we pay for the fence, others will expect for us to pay for half of theirs, but the situation is not the same! Does anyone have any experience with this? This is in Georgia, and there are no specific laws in GA about fences, and our bylaws are not specific to common area property along owner property, only owner to owner property. Any feedback would be hugely in my response to the ARC

Thank you for the questions...
1. The covenants do not say anything specific about this fence, just that shared fences are considered party fences and the cost of maintenance, replacement should be shared
2. We do have an easement as this is on one of our neighborhood lakes, but that only covers the easement from the lake up to the bottom of the fence line, where no fencing is allowed.
3. The HOA does benefit in that it blocks view of the clubhouse use from the homes in the neighborhood, keeps kids from wandering away from the playground, and it’s a terrible eyesore that would great benefit the beautification of the common spaces with its replacement.

Let me know if you have any other questions, again, new at this so I’ll take all the questions that may possibly be thrown at me.

Thank you for the feedback. I am looking for the survey information which was conducted according to those opposed to the shared cost. I get a lot of "there is history there" and "this has been done already" but then fail to provide actual proof. Being the HOA president is the stupidest job, EVER. Hah. Thanks again.

That's a private fence that is the obligation of the homeowner. This is an easy one.

Who built the fence?

Sounds pretty clearly to be a private fence - for several reasons.