Are the lone pairs in water equivalent?

I've read that the oxygen atom in water is $\mathrm{sp^2}$ hybridized, such that one of the oxygen lone pairs should be in an $\mathrm{sp^2}$ orbital and the other should be in a pure p atomic orbital.

First, am I correct about the lone pairs being non-equivalent?

Second, if so, does this have any significance in actual physical systems (i.e. is it a measurable phenomenon), and what is the approximate energy difference between the pairs of electrons?

Lastly, if it turns out the lone pairs are actually inequivalent, can this be reconciled with the traditional explanation (due to VSEPR theory) that oxygen is $\mathrm{sp^3}$ and the lone pairs are equivalent?