I think the final PER-SYD is not legal. Only one transcon allowed in Australia (except in transit to JNB). Also I've always thought the RTW ends once you return to the city of origin (extra legs in the country of origin is allowed). You could do ADL-SYD-PER as the final part, I suppose. The rest of it looks fine, you meet the middle east restriction of only 2 legs to LHR.