Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?

Question

A version of Homotopy Hypothesis says that the Fundamental $n$-grupoids model Homotopy $n$-types... and if we continue upto $\infty$, then the Fundamental $\infty$- groupoids or Kan Complexes model Topological spaces upto Weak Homotopy.

Now in Higher Gauge Theory there is a notion of Path Groupoid of a smooth space https://ncatlab.org/nlab/show/path+groupoid which is a refinement of fundamental groupoid of a topological space and which remembers more information(Thin homotopy class) than the fundamental Groupoid (homotopy class). Analogous to the notion of Higher Fundamental Groupoids of a topological space one can define higher Path Groupoids of a smooth space https://ncatlab.org/nlab/show/path+n-groupoid. Also one can go upto $\infty$ and can analogously define Path $\infty$-Groupoid of a smooth space https://ncatlab.org/nlab/show/path+infinity-groupoid.

My Questions are the following:

1. What do Path $\infty$-Groupoid of a smooth space models?
2. Is there an analogous smooth or geometric version of Homotopy Hypothesis which deals with Path $\infty$-Groupoid of a smooth space?

Now according to Remark 4.3 in https://ncatlab.org/nlab/show/diffeological+space#OnTopologicalHomotopyTypeAndDiffeologicalShape one can present Path $\infty$-Groupoid of a smooth space as a diffeological singular simplicial set. Also they mentioned that the diffeological singular simplicial set $Sing_{diff}(X)$ of a smooth space $X$ is equivalent to the "Shape" of $X$. (Though I could not understand clearly what they actually meant by "shape" of $X$).

Now according to https://ncatlab.org/nlab/show/smooth+infinity-stack#in_terms_of_groupoids_internal_to_smooth_spaces an $\infty$-groupoid internal to diffeological spaces models Smooth $\infty$-stack. Now I guess Path $\infty$-Groupoid of a smooth space is also an $\infty$-groupoid internal to diffeological spaces. So I am expecting Path $\infty$-Groupoid to model Smooth $\infty$-stack.

So in that case I am expecting the answer to my questions is:

Path $\infty$-Groupoid of a smooth space $X$ models Smooth $\infty$-stack over the site $O(X)$ and a geometric/smooth version of Homotopy Hypothesis will say that the Path $\infty$-Groupoid of $X$ is same as Smooth $\infty$-stack over $O(X)$ which is same as a diffeological singular simplicial set $Sing_{diff}(X)$.

But still my answer is incomplete in the sense that I still do not know completely what extra information does the Path $\infty$-Groupoid of $X$ captures about the smooth space $X$ other than the homotopical information of the underlying topological space of $X$.

Also I am feeling little positive about my expectations because of the following reason:

Roughly the usual Homotopy Hypothesis makes sense because $Top_{Quillen}$ is Quillen Equivalent to $sSets_{Quillen}$ via the total singular complex functor $Sing$ and the geometric realization functor $|-|$. Now according to https://ncatlab.org/nlab/show/model+structure+on+diffeological+spaces I am guessing that we may have different model structures on the category of diffeological spaces(Diff). Then for my claim of "Smooth or Geometric Homotopy Hypotheis" to be more meaningful I should expect a Quillen equivalence of Model categories between Diff(with some Model structure) and sSets(with some Model Structure). Now according to https://ncatlab.org/nlab/show/model+structure+on+diffeological+spaces#details there exists a Quillen equivalence between Diff(with some appropriate model structure) and $Top_{Quillen}$.(Though it is mentioned in the same page that there may exist some gap in the proof of this Quillen Equivalence). Now If I assume for the time being that there is no gap in the proof then in that case we get that Diff(with appropriate Model Structure) is Quillen equivalent to $sSets_{Quillen}$(because $sSets_{Quillen}$ is Quillen equivalent to $Top_{Quillen}$). But I am not sure whether this Quillen equivalence is given via $Sing_{diff}$ and $|-|$ or not as mentioned in https://ncatlab.org/nlab/show/diffeological+space#OnTopologicalHomotopyTypeAndDiffeologicalShape.

If this is not the case then I can't say anything. But if it is the case then this is what I expected. Even if it is true still this Quillen Equivalence is not a sufficient condition for my claim of Smooth or Geometric Homotopy Hypotheis" to hold true but definitely it can give a positive feeling about my claim.

Am I misunderstanding anything? Or are my expectations not making much sense?

I am not an expert, so apology in advance if I sound stupid.