I recommend testing the headphone jack with a pair of headphones if not already done. This would verify that the sound care is physically working.
Also, what happens if you uninstall the conexant sound card from device manager and then reboot? The card can be uninstalled by right-clicking the sound card, selecting uninstall (check the software box if it is an option), and rebooting when finished. Once the system reboots, it will use a generic Windows driver for the sound card. If it works, then you can stop there or install the official HP driver. If it doesn't work, check device manager to ensure the reinstall was successful and then try the HP driver.
Also, what happens if you uninstall the conexant sound card from device manager and then reboot? The card can be uninstalled by right-clicking the sound card, selecting uninstall (check the software box if it is an option), and rebooting when finished. Once the system reboots, it will use a generic Windows driver for the sound card. If it works, then you can stop there or install the official HP driver. If it doesn't work, check device manager to ensure the reinstall was successful and then try the HP driver.