P3223QE - USB-C Video Debian Support

Hello,

I received a brand-new DELL P3223QE monitor, which I bought, among other things, for its USB-C Hub (USB + Ethernet + Video) so I wouldn't need a docking station.
Except, it tells me "No USB-C signal from your device".

Summary of the situation:
DELL XPS13 9350 running Debian 11 (Librazik).

With another Windows PC and a direct USB-C connection, the screen displays perfectly.
With this PC + Dell 7-in-1 USB-C Multiport Adapter - DA310, the screen displays perfectly.

Here's the lsusb result in different configurations (PC only / PC + P2332QE / PC + DA310 + P2332QE / PC + DA310)

os@Librazik:~$lsusb
Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 001 Device 005: ID 0c45:670c Microdia Integrated Webcam HD
Bus 001 Device 004: ID 04f3:20d0 Elan Microelectronics Corp. Touchscreen
Bus 001 Device 003: ID 0a5c:6412 Broadcom Corp. BCM2045A0
Bus 001 Device 002: ID 046d:c52b Logitech, Inc. Unifying Receiver
Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
os@Librazik:~$ lsusb
Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 003 Device 003: ID 0424:284c Microchip Technology, Inc. (formerly SMSC) Hub Feature Controller
Bus 003 Device 004: ID 0bda:8153 Realtek Semiconductor Corp. RTL8153 Gigabit Ethernet Adapter
Bus 003 Device 002: ID 0424:2806 Microchip Technology, Inc. (formerly SMSC) USB2806 Smart Hub
Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 001 Device 005: ID 0c45:670c Microdia Integrated Webcam HD
Bus 001 Device 004: ID 04f3:20d0 Elan Microelectronics Corp. Touchscreen
Bus 001 Device 003: ID 0a5c:6412 Broadcom Corp. BCM2045A0
Bus 001 Device 002: ID 046d:c52b Logitech, Inc. Unifying Receiver
Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
os@Librazik:~$ lsusb
Bus 004 Device 003: ID 0bda:8153 Realtek Semiconductor Corp. RTL8153 Gigabit Ethernet Adapter
Bus 004 Device 002: ID 1d5c:5500 Fresco Logic Frescologic USB3.1Gen2 HUB
Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 003 Device 004: ID 413c:c010 Dell Computer Corp. Dell DA310
Bus 003 Device 005: ID 0424:284c Microchip Technology, Inc. (formerly SMSC) Hub Feature Controller
Bus 003 Device 006: ID 0bda:8153 Realtek Semiconductor Corp. RTL8153 Gigabit Ethernet Adapter
Bus 003 Device 003: ID 0424:2806 Microchip Technology, Inc. (formerly SMSC) USB2806 Smart Hub
Bus 003 Device 002: ID 1d5c:5510 Fresco Logic Frescologic USB2.0 HUB
Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 001 Device 005: ID 0c45:670c Microdia Integrated Webcam HD
Bus 001 Device 004: ID 04f3:20d0 Elan Microelectronics Corp. Touchscreen
Bus 001 Device 003: ID 0a5c:6412 Broadcom Corp. BCM2045A0
Bus 001 Device 002: ID 046d:c52b Logitech, Inc. Unifying Receiver
Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
os@Librazik:~$ lsusb
Bus 004 Device 003: ID 0bda:8153 Realtek Semiconductor Corp. RTL8153 Gigabit Ethernet Adapter
Bus 004 Device 002: ID 1d5c:5500 Fresco Logic Frescologic USB3.1Gen2 HUB
Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 003 Device 003: ID 413c:c010 Dell Computer Corp. Dell DA310
Bus 003 Device 002: ID 1d5c:5510 Fresco Logic Frescologic USB2.0 HUB
Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 001 Device 005: ID 0c45:670c Microdia Integrated Webcam HD
Bus 001 Device 004: ID 04f3:20d0 Elan Microelectronics Corp. Touchscreen
Bus 001 Device 003: ID 0a5c:6412 Broadcom Corp. BCM2045A0
Bus 001 Device 002: ID 046d:c52b Logitech, Inc. Unifying Receiver
Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub

What do you think? Can it work? How to make it work?

Thanks in advance for your suggestions.