The nLab page on $\infty$-categories splits the known definitions of $\infty$-categories into two types:

• Algebraic $\infty$-categories, in which composition is expressed "externally", e.g. as a some kind of map "$X_1\times_{X_0}X_1\to X_1$";
• Geometric $\infty$-categories, in which composition is expressed "internally", being defined by means of existential assertions with unicity of composition holding only up to a contractible space of choices.

From what I understand (i.e. very little), the current known definitions of globular $\infty$-categories are all algebraic, including e.g. Batanin $\infty$-categories and Grothendieck–Maltsiniotis $\infty$-categories.

Part of the problem, as I understand it, is that the (reflexive or not) globe category $\mathbb{G}$ is not a test category. From what I gather, this means that we don't have a straightforward way to put a model structure on $\mathsf{Fun}(\mathbb{G}^\mathrm{op},\mathsf{Set})$ modelling the homotopy theory of $\infty$-groupoids, as described in the nLab page model structure on presheaves over a test category.

1. Does this rule out $\mathsf{Fun}(\mathbb{G}^\mathrm{op},\mathsf{Set})$ admitting any model structure modelling $\infty$-categories or $\infty$-groupoids?
2. Even if this is so, would it possible (at least in principle) to nevertheless develop a "geometric definition"¹ of globular $\infty$-categories without making use of model categories altogether², and then show that the resulting theory is equivalent to the usual homotopy theory of $\infty$-categories in some appropriate sense (other than "Quillen equivalent")?

¹Here I have in mind something like finding a family of globular sets that fulfills in the globular theory a similar role as to what the horns $\Lambda^n_k$ do for Kan complexes, quasicategories, and $(\infty,2)$-categories.

²I've heard that we have a similar situation for Lurie's new model of $(\infty,2)$-categories, in the sense that there isn't a model structure on $\mathsf{sSet}$ recovering them as its fibrant objects. (It is this assertion that made me wonder about this question in the first place, but is it indeed true?)