Does alcohol hydrogen in the enol tautomer of a β-dicarbonyl have sp² character?

My reasoning for the somewhat implausible "$\mathrm{sp^2}$" character of the hydrogen between these two oxygens on the enol structure of the right comes from recognizing that if it were, then we would have 6 $\mathrm{sp^2}$ atoms with 6 π-electrons, so maybe that extra bit of instability from having a higher orbital on hydrogen could be sort of taken care of by being stable like an aromatic compound and forming a hydrogen bond.


So how could I go about figuring out if this is true, either with MO theory or something else?



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> [![dicarbonyl](https://i.stack.imgur.com/ufSEO.png)](https://i.stack.imgur.com/ufSEO.png)
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(source: [ucdavis.edu](http://chemwiki.ucdavis.edu/@api/deki/files/2937/image011.png?revision=1)) 


