How to get count of new values by months?

Question

I have a collection which includes documents with this structure:

{
    car: carId,
    seller: sellerId,
    buyer: buyerId,
    timestamp: timestamp
}

I want to extract unique new(first time owned a car) car owners count by monthly.

So expected result would be like:

[
    {
        date: 2022-01-01T00:00:00.000+00:00,
        newCarOwners: 28
    },
    {
        date: 2022-02-01T00:00:00.000+00:00,
        newCarOwners: 18
    },
    {
        date: 2022-03-01T00:00:00.000+00:00,
        newCarOwners: 10
    }
    ...
]

I think I can achieve it by grouping documents by month and carId, then find owners (last transaction's buyerId for each carId) by month. Then exclude earlier months'(cumulative) owners.

$setWindowFields might be useful but I couldn't find a way to exclude earlier months' owners.

Right now my pipeline looks like this:

[
    {
        $set: {
            month: {
                $dateTrunc: {
                    date: {
                        $toDate: '$timestamp'
                    },
                    unit: 'month',
                    binSize: 1,
                    timezone: 'GMT',
                }
            }
        }
    }, {
        $group: {
            _id: {
                month: '$month',
                carId: '$carId'
            },
            doc: {
                $max: {
                    ts: '$timestamp',
                    buyer: '$buyer'
                }
            }
        }
    }, {
        $group: {
            _id: {
                month: '$_id.month'
            },
            owners: {
                $addToSet: '$doc.buyer'
            }
        }
    }
]

I'm open for any idea/solution, it could be some other way than mine too.

Thanks.