Chrome sync pauses when quitting and opening Chrome

The following guide linked helps in keeping sync enabled in Chrome:

Resume sync

If you signed out of a Google service, such as Gmail, Chrome will pause sync. To turn sync back on, sign in to your Google Account:

On your computer, open Chrome.
At the top right, click Paused Sign in again.
Sign in to your Google Account.

Keep sync on when you quit or restart Chrome

If sync turns off every time that you close Chrome, you can change your settings.

On your computer, open Chrome.
At the top right, click More Settings.
At the bottom, click Advanced Site settings Cookies.
Turn off Clear cookies and site data when you quit Chrome.