The moment she mentions her Dream, she'll turn the tables on Rand.

First and foremost, that will cement the Aes Sedai behind her. They ignored her Dream and her interpretation of it once before, and paid greatly for it. They will not do so again.

The Wise Ones will completely support her, since they know, more than anyone else, that a Dreamer's interpretation is what counts.

With the WO and AS solidly behind Egwene, the Windfinders and the Kin will fall in line too. Already, you have what very neatly resembles the Fateful Concord.

Nynaeve might now disagree, but I'm unsure she'll disregard Egwene's dream. The true mystery in all this is why Egwene didn't lead her arguments against Rand with her Dream. Maybe she means to keep to the end for dramatic effect. Brandon screwed up by not showing her reasoning.

With all, or most, of the female channelers supporting Egwene, the nations will almost certainly go in her direction too.

And this is supported by Egwene's previous Dream, where Rand confronts a bunch of women around Egwene, and a Seanchan is one of them:

Rand confronting her, and the women with her, and one of them was a Seanchan.

It seems clear to me, that far from losing her power, she will consolidate it when she confronts Rand. A Latra-Lews type confrontation is brewing, and as in the past, I think both Rand and Egwene have a piece of the puzzle right.