The Hyperflash is supported as written in the LS2088A RM, 21.11.1.1 Read Command (Spansion Hyperflash) and 21.11.1.3 Word Program (Spansion Hyperflash).
The LS2088A does not provide differential clock on the QSPI interface.
Considering that is possible to use 3V flash and voltage shifters 1.8V to 3V on all signals of the bus, but highest possible data rate (333 MBs) cannot be achieved.
Please refer to the LS2088A Data Sheet to calculate maximum supported SCK frequency because it depends on the connected SPI Flash timing parameters.