Filter DataFrame by maximum timestamp column

Question

How can I filter this Pandas DataFrame so that I only get "VALUE" and "DATE" relating to the most recent "PUBLISH_TIME"?

Current DataFrame:

PUBLISH_TIME            DATE        VALUE
0   2022-01-04 00:52:30 2022-01-04  490
1   2022-01-04 10:01:45 2022-01-04  503
2   2022-01-04 10:09:04 2022-01-04  504
3   2022-01-05 00:44:16 2022-01-05  513
4   2022-01-05 10:18:44 2022-01-05  527
5   2022-01-06 01:03:16 2022-01-06  527
6   2022-01-06 10:09:08 2022-01-06  520

What I want:

DATE        VALUE
2022-01-04  490
2022-01-05  513
2022-01-05  527

Code:

import pandas as pd

x = {
    "PUBLISH_TIME": [
        "2022-01-04 00:52:30", 
        "2022-01-04 10:01:45",
        "2022-01-04 10:09:04",
        "2022-01-05 00:44:16",
        "2022-01-05 10:18:44",
        "2022-01-06 01:03:16",
        "2022-01-06 10:09:08"
        ],
     "DATE": [
         "2022-01-04",
         "2022-01-04",
         "2022-01-04",
         "2022-01-05",
         "2022-01-05",
         "2022-01-06",
         "2022-01-06"
     ],
     "VALUE": [
         490,
         503,
         504,
         513,
         527,
         527,
         520
     ]
}  
     
df = pd.DataFrame(x)

I have tried df.groupby('PUBLISH_TIME').apply(max)[['DATE', 'VALUE']] however this does not return the desired result.