Bloomberg, in its documentation, explains that it calculates the short rate volatility for its Hull White implementation by multiplying the e.g. 10y IRS rate (divided by 100) by the 10y cap vol.
I must be missing something obvious.
Finally got a response in terms of more documentation, and a half sentence that mentions lognormal vs normal, which I'm assuming means they are saying that the cap vol is a Black (i.e. lognormal) volatility of dr/r, but we want a normal volatility for the Hull-White model of dr (which they use). So multiply by r both sides.
Which is, I suppose, approximately the right idea? Except why the 10y cap vol and 10y rate, when the Hull-White model is a one-factor model of the short rate; the 10y rate is not the short rate, even if we assume the volatility is constant and equal to the value at 10y.