Since gmail introduced the new split inboxes open rates have gone down, fact. If you want to read more into it have a look at this blog post from mailchimp. It’s not all bad, fewer emails are ending up in spam as a result of the new sorting by gmail, see what marketing land had to say about this issue on their blog.
Only via the gmail web inbox and the gmail app are these emails being sorted, if you are using another bit of software on your computer or phone then everything is still coming to the one generic inbox.
Should you try and convince your users to move you back into your primary inbox? I don’t see the need, the example on the right is one I personally just deleted.
The new inboxes are necessarily all bad for email marketers, it can actually give consumers an opportunity to go through these types of emails when they actually have time to do so and if you are providing relevant, engaging content then your loyal fans could me more inclined to act on your promotional emails.