@Stéphane Lalancette, Thanks for posting in Q&A. The Config Refresh feature in MDM (Mobile Device Management) is designed to improve the security and compliance of managed devices by allowing more frequent policy refreshes.
Config Refresh does support offline functionality, meaning it allows us to bring our device back to a desired state (policies) even when it is offline. We can configure a cadence to let Windows reapply the Intune-configured policies to the device. Config Refresh will query the policymanager\providers for all Intune configured policies. Once it knows which policies were configured and how, it will tag them, delete them, and set them once again. With it, the policies are refreshed. Sounds like the configuration was just refreshed! Here is a blog with more details to let you understand,
https://call4cloud.nl/config-refresh-providers-cache/
Note: None-Microsoft link, just for your reference.
For the scheduled task, based on my testing, it sets "Start only if the following network connection is available" as any connection in Conditions by default when we configure Config Refresh policy. Based on my researching, I find when it is set any connection, it checks for the availability of a network connection, which could also be a local network without internet access. if there's local connection on the testing devices, the task can start. If there's no network connection available on the device, it may affect, you can click the task and choose Properties to uncheck this condition to make it start.
Hope the above information can help.
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