I don't see this fitting any of the OW Circle or RTW fares.

However, there is an ambiguity in the rules which a number of people have used to get something like this ticketed:

CAI-LHR-HKG-(SIN-HKG-)-SYD-AKL-SYD-(x)HKG-<N. America>-LHR-CAI

In this case this is a xONE4 OneWorldExplorer ticket and you can take up to a total of 20 flight segments with a maximum of 4 in any continent (6 in N America). Only two stopovers allowed in Europe.

The prices are in the sticky thread at the top of this forum. Cairo is generally only worth the effort for J or F.