  The $k$-server conjecture, first posed by Manasse, McGeoch and Sleator in
1988, states that a $k$-competitive deterministic algorithm for the $k$-server
problem exists. It is conjectured that the work function algorithm (WFA)
achieves this guarantee, a multi-purpose algorithm with applications to various
online problems. This has been shown for several special cases: $k=2$,
$(k+1)$-point metrics, $(k+2)$-point metrics, the line metric, weighted star
metrics, and $k=3$ in the Manhattan plane.
  The known proofs of these results are based on potential functions tied to
each particular special case, thus requiring six different potential functions
for the six cases. We present a single potential function proving
$k$-competitiveness of WFA for all these cases. We also use this potential to
show $k$-competitiveness of WFA on multiray spaces and for $k=3$ on trees.
While the DoubleCoverage algorithm was known to be $k$-competitive for these
latter cases, it has been open for WFA. Our potential captures a type of lazy
adversary and thus shows that in all settled cases, the worst-case adversary is
lazy. Chrobak and Larmore conjectured in 1992 that a potential capturing the
lazy adversary would resolve the $k$-server conjecture.
  To our major surprise, this is not the case, as we show (using connections to
the $k$-taxi problem) that our potential fails for three servers on the circle.
Thus, our potential highlights laziness of the adversary as a fundamental
property that is shared by all settled cases but violated in general. On the
one hand, this weakens our confidence in the validity of the $k$-server
conjecture. On the other hand, if the $k$-server conjecture holds, then we
believe it can be proved by a variant of our potential.
