The concept of a "higher dimensional category" is still an active area of research and remains a topic of ongoing investigation in mathematics. As such, the notion of a "higher dimensional category of all higher dimensional categories" is still an open question and does not have a definitive answer at this time.
In higher category theory, mathematicians study structures that generalize the notion of categories, such as 2-categories, 3-categories, and so on. These higher-dimensional structures allow for more intricate relationships and compositions between objects and morphisms.
There are various approaches and frameworks for developing higher dimensional category theory, such as simplicial categories, quasicategories, and higher topos theory. These frameworks aim to capture the structure and behavior of higher dimensional categories.
However, constructing a higher dimensional category that encompasses all other higher dimensional categories is a challenging task. It would require establishing a foundational framework that encompasses and organizes all possible higher dimensional structures. Currently, mathematicians are actively researching and developing theories to address this question, but a definitive answer or a universally accepted higher dimensional category of all higher dimensional categories has not been established.
It's important to note that the field of higher category theory is still evolving, and new insights and developments may emerge in the future that shed more light on this question.