In the mean time the kernel for stable 23.05.2 reached 5.15, so AQC111U should in theory be working. Some posts state working connections (e.g. Updating OpenWrt when you need a USB Ethernet driver - #6 by frollic).
I tried it on kernel 5.15 (Turris Omnia 6.4.4, based upon OpenWrt 21.02) and it doesn't work. Does someone know how to troubleshoot that situation?
root@turris:~# lsusb
Bus 004 Device 001: ID 1d6b:0002 Linux 5.15.135 xhci-hcd xHCI Host Controller
Bus 005 Device 003: ID 1c04:0015 QNAP▒ Pacific▒
[ ... ]
[ 2768.079584] usb 5-1: new SuperSpeed USB device number 3 using xhci-hcd
[ 2991.436031] kmodloader: loading kernel modules from /etc/modules.d/*
[ 2991.444759] usbcore: registered new interface driver aqc111
[ 2991.451052] kmodloader: done loading kernel modules from /etc/modules.d/*
[ 2991.464188] aqc111 5-1:1.0 eth3: register 'aqc111' at usb-f10f8000.usb3-1, QNAP QNA-UC5G1T USB to 5GbE Adapter, [...MAC...]
[ 2991.678116] br-lan: port 1(eth3) entered blocking state
[ 2991.683381] br-lan: port 1(eth3) entered disabled state
[ 2991.688742] device eth3 entered promiscuous mode
[ 2991.693456] br-lan: port 1(eth3) entered blocking state
[ 2991.698703] br-lan: port 1(eth3) entered forwarding state
[ 2991.704476] br-lan: port 1(eth3) entered disabled state
When connecting a ethernet cable, the LEDS start to blink, but there is no connectivity at all. The adapter works perfectly well when used with Windows.