Adding URP to a project does not respect Camera's Target Eye value

Steps to reproduce:
1. Open the attached user's project "VR Spectator.zip"
2. In the Hierarchy, select the "VR Spectator Camera" GameObject
3. In the Inspector, under Camera -> Output (if it's available), observe the "Target Eye" property being set to "None"
4. Open Project Settings -> Graphics and add a URP Render Pipeline Asset
5. Repeat step 3 and observe that the "Target Eye" property is now set to "Both"

Expected Result: Camera's Target Eye value remains the same no matter if URP is added or not
Actual Result: Adding URP to a project does not respect Camera's Target Eye value

Reproducible with: 2021.3.33f1, 2022.3.17f1, 2023.2.5f1, 2023.3.0b1

Testing Environment: Windows 11
Not reproducible on: No other environment tested