Ease of use – button group to filter the time range without a selected default value

I am currently working on a timeline component. By default, the activities / events for the day are displayed in this timeline.

Users can select predefined time ranges such as day, week, month, etc. It also has a date picker that allows the user to select a custom date range.

The problem I'm facing here is that the button group's default time range is "Today". However, if the user selects a custom date range value, no value is selected for the button group.

I know that button groups are similar to radio buttons, and it's okay not to select a value for option groups by default if an accepted default is not possible. (How payment options)

But does that work in Button Groups / Segmented Control?

Is there an established pattern for this scenario?

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