I am looking for references concerning the following facts, which I believe to be true:

In a Kan complex $K$, can I use "empty cubes" $\partial(\Delta^1)^{(n+1)}$ with the vertex $(0,0,\ldots,0)$ at the basepoint (under the obvious notation $0$ and $1$ for the two $0$-simplices of $\Delta^1$) as representatives for the simplicial homotopy groups of $K$?

More precisely, I would like to have an "algorithm" (i.e. a precise order in which to do the homotopies) to homotope the empty cube to to a single $n$-simplex with the boundary mapping to the basepoint. Moreover, I would also like to have the extension to the whole cube: If the empty cube can be filled then the homotopy class should be trivial, and I would like an extension of the homotopy giving the corresponding homotopy of the $n$-simplex to the basepoint.

*Note:* I want to do all of this in simplicial sets, so no cubical sets. This question is related but works with a different representation of homotopy classes (and also does not assume $K$ to be Kan). It does not answer my question in a satisfactory way.

*Note 2:* I am aware that one of the reasons we work with simplices is that for many things they behave "better" than cubes. Unfortunately, I would need what I detailed above for a nice application that I'm afraid would become horribly complicated if I were to use the standard definition of simplicial homotopy groups.