Replacing column values by aggregating on groups

Suppose I have the following dataframe df:

   id   x    y        timestamp
   1   32   30        1031
   1   4    105       1035
   1   8    110       1050
   2   18   10        1500
   2   40   20        1550
   2   80   10        1450
....


import pandas as pd
import numpy as np
df = pd.DataFrame(np.array([[1,1,1,2,2,2], [32,4,8,18,40,80], [30,105,110,10,20,10], [1031,1035,1050,1500,1550,1450]])).T
df.columns = ['id', 'x', 'y', 'timestamp']

I now have the following code:

df= df.groupby(["id"]).agg({
    'timestamp': lambda x: x.max() - x.min(),
    'x': 'mean',
    'y': 'mean'
}).reset_index()

Unfortunately, this is not what I want. I want the following resulting dataframe:

id    x     y       timestamp
-----------------------------
1    32     30      19
1    4      105     19
1    8      110     19
2    10     10      100
2    40     20      100
2    80     10      100
....

This means that the timestamp column should be replaced with the max-min value for each group (but I don't want to aggregate the whole group into a single value).

How can this be done?