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I believe the investment limit (>2000 gold) makes your family wealthy enough for you to ask for some gold without losing h/p.
I've had more than two thousand gold (just above, but enough to give without going below), and asked for some for the first time on a particular character. Lost 1 or 2 prestige.
Someone want to start a wiki page for this? It might be useful material, and it would help to accumulate the findings we make, so interested readers wouldn't have to dig through this thread, if it ever gets long enough.