The KNP asks the maximum number of $n$-spheres can touch a given $n$-sphere in the center. Are these $n$-spheres required to touch each other as well?
In that case we can decompose the structure in a particular dimension (i.e. the $A_n$ lattices) into a bunch of regular $n$-simplexes.
In KNP the other spheres are not required to touch each other, only the center sphere must touch every other sphere.
If the case was that other spheres are required to touch each other, then YES they can be considered as $n$-simplexes.
Also the simplexes are well defined and the number of spheres forming the $n$-simplex is given by $n+1$ but there is ambiguity in finding the number of $n$-spheres in a lot of dimensions.