The problem is quite general not specific of my Dell hardware either, see below a few related web pages, none reporting a convincing solution. What I describe is completely reproducible under both 6.08 and 6.11 kernels (the latter is used in the attached output).

My Dell XPS 13 9340 detects the external monitor correctly only when [condition 1] I power on the laptop, hit F2 and start the UEFI BIOS, exit from BIOS (with of without changing it) and boot with the HDMI cable already connected. Hardware and software details are below. Grub selecting Windows boot the external monitor is always recognized like this.

If I disconnect the cable and connect it again, or suspend the pc (to S3 state), and thereafter, any time I reboot the laptop with or without connected cable, the external monitor is not recognized any more [condition 2]. This is particularly crippling since it practically prevents the use of external monitors and overhead projectors.

I show below the output of a few diagnostics in the two conditions.

In condition I

> xrandr --listactivemonitors

> Monitors: 2
>  0: +*eDP-1 1920/290x1200/180+0+1440  eDP-1
>  1: +DP-6 2560/700x1440/390+0+0  DP-6

> inxi -GSaz --vs
>inxi 3.3.34-00 (2024-04-13)
>System:
>  Kernel: 6.11.0-17-generic arch: x86_64 bits: 64 compiler: gcc v: 13.3.0
>    clocksource: tsc avail: acpi_pm
>    parameters: BOOT_IMAGE=/boot/vmlinuz-6.11.0-17-generic
>    root=UUID=19de882b-011d-46dc-bb01-3fa016e2504a ro
>    resume=UUID=4aab95f5-8481-4a4e-b7ce-dcdcec31ee83
>  Desktop: GNOME v: 46.0 tk: GTK v: 3.24.41 wm: gnome-shell
>    tools: gsd-screensaver-proxy dm: GDM3 v: 46.2 Distro: Ubuntu 24.04.2 LTS
>    (Noble Numbat)
>Graphics:
>  Device-1: Intel Meteor Lake-P [Intel Arc Graphics] vendor: Dell driver: i915
>    v: kernel alternate: xe arch: Gen-13 process: Intel 4 (7nm+) built: 2023+
>    ports: active: DP-6,eDP-1 empty: DP-1, DP-2, DP-3, DP-4, DP-5
>    bus-ID: 00:02.0 chip-ID: 8086:7d55 class-ID: 0300
>  Display: wayland server: X.org v: 1.21.1.11 with: Xwayland v: 23.2.6
>    compositor: gnome-shell driver: gpu: i915 display-ID: 0
>  Monitor-1: DP-6 model: Lenovo LEN T32q-27 serial: <filter> built: 2022
>    res: 2560x1440 dpi: 93 gamma: 1.2 size: 698x393mm (27.48x15.47")
>    diag: 801mm (31.5") ratio: 16:9 modes: max: 2560x1440 min: 720x400
>  Monitor-2: eDP-1 model: Sharp 0x1593 built: 2023 res: 1920x1200 dpi: 169
>    gamma: 1.2 size: 288x180mm (11.34x7.09") diag: 340mm (13.4") ratio: 16:10
>    modes: 1920x1200
>  API: EGL v: 1.5 hw: drv: intel iris platforms: device: 0 drv: iris
>    device: 1 drv: swrast gbm: drv: iris surfaceless: drv: iris wayland:
>    drv: iris x11: drv: iris
>  API: OpenGL v: 4.6 compat-v: 4.5 vendor: intel v: N/A glx-v: 1.4
>    direct-render: yes renderer: Mesa Intel Arc Graphics (MTL)
>    device-ID: 8086:7d55 memory: 15.04 GiB unified: yes display-ID: :0.0

> cat /sys/class/drm/card1/card1-eDP-1/status 
>  connected
> cat /sys/class/drm/card1/card1-DP-6/status 
>  connected
> cat /sys/class/drm/card1/card1-DP-1/status 
> ...
> cat /sys/class/drm/card1/card1-DP-5/status 
>  disconnected

The Menu button of the LG external monitor is active and reports

> Input Signal:HDMI Resolution: 2560x1440 Refresh: 60Hz

In [condition 2]

> xrandr --listactivemonitors
>Monitors: 1
>  0: +*eDP-1 1920/290x1200/180+0+1440  eDP-1

> cat /sys/class/drm/card1/card1-eDP-1/status 
>  connected
> cat /sys/class/drm/card1/card1-DP-1/status 
> ...
> cat /sys/class/drm/card1/card1-DP-4/status 
>  disconnected

while 
> cat /sys/class/drm/card1/card1-DP-5/
> cat: /sys/class/drm/card1/card1-DP-5/status: No such file or directory
> cat /sys/class/drm/card1/card1-DP-6/
> cat: /sys/class/drm/card1/card1-DP-6/status: No such file or directory

The Menu button of the LG external monitor hungs, it is not possible to move the cursor in the menu display and the report is incomplete

> Input Signal:HDMI [no Resolution and Refresh]

Similar situations are described for other distros (Arch, Gentoo, Manjaro) and hardware (Lenovo)