Kolchin's problems

Kolchin's problems are a set of unsolved problems in differential algebra, outlined by Ellis Kolchin at the International Congress of Mathematicians in 1966 (Moscow)

Kolchin Catenary Conjecture
The Kolchin Catenary Conjecture is a fundamental open problem in differential algebra related to dimension theory.

Statement
"Let $$ \Sigma $$ be a differential algebraic variety of dimension $$ d $$ By a long gap chain we mean a chain of irreducible differential subvarieties $$ \Sigma_0 \subset \Sigma_1 \subset \Sigma_2 \subset \cdots $$ of ordinal number length $$ \omega^m \cdot d $$."

Given an irreducible differential variety $$ \Sigma $$ of dimension $$ d > 0 $$ and an arbitrary point $$ p \in \Sigma $$, does there exist a long gap chain beginning at $$ p $$ and ending at $$ \Sigma $$?

The positive answer to this question is called the Kolchin catenary conjecture.