  Garside-theoretical solutions to the conjugacy problem in braid groups depend
on the determination of a characteristic subset of the conjugacy class of any
given braid, e.g. the sliding circuit set. It is conjectured that, among rigid
braids with a fixed number of strands, the size of this set is bounded by a
polynomial in the length of the braids. In this paper we suggest a more precise
bound: for rigid braids with $N$ strands and of Garside length $L$, the sliding
circuit set should have at most $C\cdot L^{N-2}$ elements, for some constant
$C$. We construct a family of braids which realise this potential worst case.
Our example braids suggest that having a large sliding circuit set is a
geometric property of braids, as our examples have multiple subsurfaces with
large subsurface projection; thus they are "almost reducible" in multiple ways,
and act on the curve graph with small translation distance.
