Edit: After half day, my wife’s Mac started having the same problem. I have then applied the solution that I found in the Apple
How to Use the VIP Inbox
In addition to your regular inbox, the Mail app on your iOS device will also come with its own VIP inbox for all those people who mean more to you. Setting a contact as a favourite means that any emails you receive from them will be filtered straight into this special inbox reserved for important people. This makes them much more easy to find quickly and in addition to this, you are also presented with the option of designating specific tones to your favourite contacts, so you can instantly tell when one of them is contacting you.
The first thing you will need to do is add contacts to the VIP list so your device recognises them. This is simple to do and whether you are using an iPhone or iPad, the process is the same either way.
First of all launch the Mail app, which you can find on the Home screen of your device. On the Mailboxes list you will see the VIP inbox below the main inbox, so tap it and then tap the blue arrow to the right of it. This will take you to the VIP list, which is where your VIP contacts will be stored once added. One of the headings is labelled Add VIP… so tap on this one. You will now be able to scroll down your contact list and search for all the lucky people you wish to add as VIPs. Tap on a name and that contact will instantly be added. You may notice that some names are grey and when you tap them they are unresponsive. This simply means you do not have that person’s email address yet, but you can at least call them and ask for it.
Once you have all your favourites in place you can then customise them slightly by giving them their own unique tones that will sound whenever they email you.
As before, locate the Mail app on your device’s Home screen and open it. Select the VIP inbox below the regular one and then tap the blue arrow on the right to go to the next menu. This time you need to ignore the contact list and look below, where you will see a button labelled VIP Alerts. Tap this one to go to the alerts menu, where you will see a few options. As you can see, Notification Centre is automatically enabled for this device but if you so wish you can flip the switch to OFF and just check emails in your own time. However, if you would like to be alerted, leave it on and choose an Alert Style. There are two forms your alert can take when it pops up and you can choose from a banner, which scrolls across the top of the screen, or an alert, which appears in the middle of the screen. In addition to this, you will also be able to toggle the Badge Alert Icon on and off and choose a New Mail Sound, which you will hear on receiving a new VIP message.