Inexistence of a Kan–Quillen model structure on globular sets (This is in a sense a follow-up to my earlier question on a geometric definition of globular $\infty$-groupoids)

We know by Scholie 8.4.14 of Cisinski's thesis that the globe category $\mathbb{G}$ is not a weak test category. Thus, there is no model structure on $\mathsf{Fun}(\mathbb{G}^\mathsf{op},\mathsf{Set})$ such that:

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*This model structure models $\infty$-groupoids;

*The cofibrations in this model structure are precisely the monomorphisms;

*The $n$-globes $G_n:=\mathrm{Hom}_{\mathbb{G}}(-,[n])$ are contractible.

Nevertheless, there could still in theory be a model structure on $\mathsf{Fun}(\mathbb{G}^\mathsf{op},\mathsf{Set})$ modelling $\infty$-groupoids, even though it might be really bad behaved, and some of the constructions in it would be very ad hoc¹.
¹In particular we would need to somehow construct a geometric realisation functor without using the abstract procedure in nLab, nerve and realization, as that recovers only those homotopy types which are wedge sums of spheres (as Simon Henry explained to me here).
Question. Is the following statement true?

*

*There exists no model structure on $\mathsf{Fun}(\mathbb{G}^\mathsf{op},\mathsf{Set})$ that is Quillen equivalent to the Kan–Quillen model structure on simplicial sets.

 A: Zhen Lin points out below that I've been way too cavalier with transferring model structures along a reflection. So the following answer is not clearly correct. I will leave this up as community wiki because I still think it addresses the spirit of the question, showing that spaces can be "modeled" in some sense by globular sets (or even just by graphs).

Contrary to my guess in the comments, the answer is no: there does exist a model structure on globular sets (reflexive or otherwise) which is Quillen equivalent to the Kan-Quillen model structure on spaces.
To see this, note that if $\mathcal A$ is a reflective subcategory of $\mathcal B$, and if $\mathcal A$ has a model structure, then the model structure transfers to $\mathcal B$, and the resulting adjunction is a Quillen equivalence.
Now, as mentioned in the comments, the category $Gph$ of graphs (reflexive or otherwise) is a reflective subcategory of $Glob$. Moreover, the category $Pos$ of posets is a reflective subcategory of $Gph$. Thus $Pos$ is reflective subcategory of $Glob$. So it will suffice to find a model structure on $Pos$ which is Quillen equivalent to topological spaces. This is proven by Raptis, by transferring the Thomason model structure on $Cat$.
