Does objectivity exists?
In the mind - no. In reality - yes.
The mind is not reality, the mind exists on top of reality. For example, if you lose a limb you might experience what is called the “phantom limb pain”. It’s coming from the mind’s imagination that the lost limb still exists and there’s pain coming from it. Since it doesn’t exist in reality, there is nothing you can do physically to ease this pain, so the condition can only be successfully treated with a mind trick using mirrors.
However, no amount of mind tricks will allow you to bring the lost limb back into existence.
The psyche is completely subjective, the physical reality of your senses is completely objective, and the psyche has the capacity to distort the senses. Once you get rid of the psyche you end up in objective reality.