Esri is still releasing ArcObjects .NET SDK for ArcGIS 10.5, so at least they are supporting it for the newest release. Additionally, the What's new for developers at 10.5 does show some minor updates or changes, and at 10.5 "the .NET and VCPP samples are available online for download from https://github.com/esri/arcobjects-sdk-community-samples ."
I think the future of ArcObjects mirrors the product(s) built upon it, i.e., ArcMap, ArcGIS Engine, etc.... Although Esri won't go as far as saying ArcMap is deprecated, it is clear the amount of development resources being put into it is fading, and fast. In terms of ArcMap/ArcCatalog, ArcGIS Engine, and similar products; I wouldn't expect any new APIs or SDKs to be developed for them. The products are mature, close to over mature, and there is no incentive to develop new APIs or SDKs for those products. Even when ArcMap is deprecated, I expect its retirement path to be a long one, so there will be some space for ArcObjects for quite some time.
The future of desktop GIS with Esri is ArcGIS Pro, and Pro might not be ready for primetime now, but each release gets it closer. Looming in the background is also the continued push by Esri into Web GIS. As more and more workflows move to Web GIS and mobile apps, the need for customizing desktop GIS will decrease. Desktop GIS will never go away, but I think organization's use of it will decrease overall as Web GIS matures.
If you have ArcObjects-based customizations and code in your organization, I would personally start planning for what's next, even if you don't transition for a couple or few more years.