There are three parts to this*:
1) We don't know whether gravity treats matter and anti-matter identically (eg whether they're identically affected by the Higgs field, or <insert gravitational theory here>)
2) We know that electrons and positrons have very similar masses, but not that they're identical.
3) In a direct annhililation, the tiny fractional differences in mass would be utterly swamped by uncertainties in the measurement (both experimental and fundamental). In the proposed experiment, if the lifetime (and hence beam length) of the positronium can be made sufficiently large, it may be possible to reduce the uncertainties to such an extent that the differences are measurable.
* I don't see how the proposed experiment could disentagle the effects of 1) and 2), but maybe that's why I can only call myself a physicist if I add the prefix 'failed'.