A server shows the error in Plesk 360: Connection error

Plesk 360

Symptoms

A new Plesk server can not be added to Plesk 360 with one of the following errors:

PLESK_ERROR: The connection to the server server.example.com cannot be established correctly. Plesk version is not supported (below 18.0.37) or Plesk is misconfigured.

PLESK_ERROR: The network connection to the server timed out

PLESK_ERROR: Unknown error: connect ECONNREFUSED 213.0.1.2:443
The following error appears under Server state in Plesk 360 for an already added server:

PLESK_ERROR: Connection error
The following message appears under Plesk 360 > Server > server_hostname:

PLESK_ERROR: The connection to the server server.example.com cannot be established correctly. The server is configured incorrectly or you have entered an incorrect address.

Please fix the error and press the button below to restore the connection

Access to the server is blocked in one of the following ways:

a firewall rule blocks connections
IP addresses of Plesk 360 are filtered by IP Restrictions for Administrative access
the API filtration rules are configured

Allow access for the necessary IP addresses (52.51.23.204, 52.213.169.7, 34.254.37.129):

in a firewall (if there is the Plesk Firewall check this article)
in Plesk > Tools & Settings > IP Access Restriction Management (the instructions might be found here)
in the section [api] inside the file panel.ini (the details are here)

Automatic check the configuration from CLI

Connect to the server via SSH
Download the script:

# curl -OL https://raw.githubusercontent.com/plesk/kb-scripts/master/360-restrictions-check/360-restrictions-check.py
Make it executable:

# chmod +x 360-restrictions-check.py
Execute the script to check the restrictions:

# ./360-restrictions-check.py

If the following error appears:

CONFIG_TEXT: /usr/bin/env: 'python': No such file or directory

Execute the script in this way:

# python3 360-restrictions-check.py