The higher the share price, the less demand there will be for shares. It's simple economics. FHFA and Treasury will have to lower the offering price to where the market clears (where there is enough demand to raise all the money they need), and that is likely to be well below $15.
Recapping at $15-20 per share just doesn't work because it leads the new buyers to having less than half the equity, in return for putting in nearly all of the capital. They have no reason to accept that.