Something deep-seated is firmly established.
The phrase can be used to refer to a tradition, an emotion, a fear — anything that's ingrained and hard to alter or remove. It comes from the verb seat, meaning "to situate," and deeply seated was originally used to describe infections or diseases that were located deep in the body and were therefore difficult to treat or remove. These days, deep-seated isn't reserved only for illnesses, but it still typically carries a negative connotation. The examples below show how frequently it's used to modify words like suspicion or grievance:
It can be perceived as a quiet way of saying no one will ever forget what happened, nor the deep‑seated and unresolved grievances. (The Guardian)
The local campaign to eradicate polio remains a source of deep-seated suspicions and fears. (New York Times)
It's not surprising that people confuse deep-seated with deep-seeded. Seeds are planted below the soil, deep in the ground. The idea of an underground seed growing roots fits with the idea of something being firmly fixed and difficult to dislodge. Unfortunately, deep-seeded is simply incorrect. The following examples show how common this mistake is:
This sleeping pack is best suited for people with dry and blemish-prone skin — as it contains "skin enhancing nutrients" for parched skin as well as bee propolis which promises to get rid of "deep seeded dirt and bacteria." (Slate)
At universities around the country, students demanded their schools confront deep-seeded racial injustices and address the question of how to provide an inclusive educational environment, free of discrimination. (The Guardian)
Imagine an elderly man sinking deeply into his favorite squashy armchair — he's comfortable and settled and he just doesn't want to get up. Anything deep-seated is similarly fixed in place and difficult to move.