My understanding is that the Type C sockets fed by a cable from a mobo Type E header can support BOTH Gen2 and Gen2x2 versions. I suspect the reason is that the Gen 2x2 header operates differently from Gen 2, but at virtually the same signal rate. Say what? Well, Gen 1 and Gen 2 mobo headers, like the older USB2 system, actually contain TWO USB ports so that the cables from these headers can feed TWO external sockets operating simultaneously and independently. A Gen 2x2 header is different. It contains only ONE 2x2 port and this can feed only one socket. It's as if the two ports of other Gen's are combined into two co-ordinated communications channels each operating at the slower speed of 10 Gb/s max, but combined able to transfer data in total at 20 Gb/s. IF you plug into that socket a device able to work only at the slower Gen 2 speed, the device and mobo controller will negotiate to run at the slower speed. That's just the common way that USB3.2 controllers operate to provide backwards compatibility.
That's just MY interpretation. But I do note that the CABLE and SOCKET from a mobo Type E header to a front panel Type C socket appear to come in only one version - no specials designated for Gen 2 only or for Gen 2x2 only. MAYBE to confirm you could ask Tech Support at the case maker whether that front socket system does support Gen 2x2.