The maximal-radius distinctness conjecture of Robinson

For $N\geq 4$, let $r_{\max}(N)$ be the maximal radius of $N$ equal spheres touching a central unit sphere. Robinson's conjecture. For all $N\geq 4$, one has

$$r_{\max}(N)<r_{\max}(N-1)$$

except possibly for $N=6,12,24,48,60,120$. This is a general Tammes-problem conjecture about strict decrease of maximal radii, with the listed exceptional values; no resolution is given in the source.