Is Minimax Linkage a Lance-Williams hierarchical clustering?
My take after a quick read of the references.
First of all, Lance and Williams's original paper mentions that their linear scheme works (and offers computational advantage) only for combinatorial strategies. Is minimax linkage such a combinatorial strategy? In other words, does it depend (linearly) on pair-wise distances? By the defintion of minimax distance it is clear that this is implausible.
It is like the difference between mean and median computation in statistics. Mean is linear while Median is non-linear. There is no linear combination that can compute median from mean (although in certain limited cases they can coincide).
Second, the authors do not mention the form (or values) of the $\alpha, \beta, \gamma$ parameters in a hypothetical Lance-Williams method for minimax linkage. But in any case they are constants and $\alpha(\cdot)$ can be constant or rational functions of the respective cluster sizes (per the original Lance-Williams reference).
$G_2$ might not be of the same cardinality in the two panels, but minimax linkage distance depends on radius of cluster not cardinality (unlike average or centroid linkage) and the two examples have same radius thus $\alpha(G_2)$ is same in both cases.
Another way to see that this is the case is a plausible variation of the proof where $G_2$ in both cases has both same radius and same cardinality but different configuration.

Maybe such a proof would make it more clear. But I will leave it at that, at this point.