I see a lot of space on 6/28 on US (some Envoy sleeper) via CLT/PHL, and on UA via IAD, on LH via BOS,EWR,ORD all to FRA.
They will all connect to that single FRA-HYD flight.

On the way back, I see 7/29 in business
EK 525 HYD-DXB
LH 639 DXB-MUC
LH 434 MUC-ORD
UA 375 ORD-MCI

longest layover is 3 hours

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https://www.ana.co.jp/asw/wws/us/e/