So I installed you plugin and this is the response/conclusion I got:
Conclusion:
Port 443 can be used to send Gmail with the Postman Gmail Extension.
Port 465 is blocked. Contact your host for a solution, such as using their local SMTP server or opening the port.
Port 25 is blocked. Contact your host for a solution, such as using their local SMTP server or opening the port.
Port 587 is blocked. Contact your host for a solution, such as using their local SMTP server or opening the port.
A Closed port indicates one or more of these issues:
Your host has placed a firewall between this site and the Internet
Your PHP configuration is preventing outbound connections
Your WordPress configuration is preventing outbound connections