Conjecture on kissing numbers of three- and four-dimensional Minkowski spheres

For real $c>1$, consider the Minkowski spheres

$${\mathbb S}^{n-1}_c:\ |x_1|^c+\cdots+|x_n|^c=1.$$

The kissing number is the maximum number of non-overlapping congruent spheres tangent to a given sphere. Conjecture on Minkowski-sphere kissing numbers. For $n=3$, the kissing number of ${\mathbb S}^{2}_c$ is $12$; for $n=4$, the kissing number of ${\mathbb S}^{3}_c$ is $24$. This extends the familiar Euclidean kissing-number values in dimensions two and three to the corresponding Minkowski spheres, while the supplied text does not state whether these assertions have been proved.