I have rather strong doubts about the accuracy of the above answer, though it is in "genuine print". So far, I haven't found anything really clear that resolves this issue in the literature.
The argument that GR "doesn't allow discontinuities in the metric doesn't seem to me to prove the point in question. I do suspect that the author of the book believes his remarks to be true, but I'm not convinced that the statements made in this FAQ follow from Einstein's field equations.
The reasons why I think the approach is wrong are a bit technical, but boil down to the fact that the only way to "thread" an electric field, or by analogy a gravitational one, is by passing a charge, or mass through it.
This is incompatible with the author's idea that it would "always be uphill" if the wormhole connected two regions of differing Newtonian gravitational potential - the author would have us believe that there must always be a field threading the wormhle in this case, it's not consistent with the idea that the only way to generate such a field is to pass an object with charge (or mass) through it.