Classification of commutative rings $R$ satisfying $\dim(R[T])=\dim(R)+1$

Question

Let $R$ be a commutative ring. Then $\dim(R[T]) \geq \dim(R)+1$. Is there a classification of those commutative rings with the property $\dim(R[T])=\dim(R)+1$? Every Noetherian commutative ring has this property, but also every $0$-dimensional commutative ring (Noetherian or not) has this property (see here). A $1$-dimensional counterexample can be found here. In case it helps for the classification, we can also add closure properties and make the property stronger (which might simplify the problem). For example, is the class of commutative rings $R$ such that (every quotient of) (every localization of) (every finitely generated commutative algebra over) $R$ satisfies this dimension equality more easy to classify? The closure under finitely generated commutative algebras includes polynomial rings in particular, so that we also demand $\dim(R[T_1,\dotsc,T_n])=\dim(R)+n$ for all $n \in \mathbb{N}$.