Bakhtiari, Lattimore and Szepesv'ari (COLT 2025) proved that Thompson sampling (TS) has Bayesian regret
O~(d5/2n) for bandit convex optimisation with convex \emph{monotone} ridge losses
f(x)=ℓ(\ipxθ), and asked whether monotonicity of the link is necessary. We give a qualitative negative answer. For every prior on
[0,1]-valued,
1-Lipschitz convex ridge losses with an arbitrary convex, possibly non-monotone, link, and for any fixed measurable selection of minimisers, exact-posterior TS has Bayesian regret
O((d+1)4dnlog(e+ndmax{1,\diamK}))=O~(d9/2n). The monotone proof relies on a single-removal John-ellipsoid dichotomy; we show by an explicit twelve-point configuration that this dichotomy fails for non-monotone links, and replace it by an
O(d2) cardinality bound for ``uninformative'' configurations. The bound uses a Boolean rounding argument: a
0-
1 matrix within
1/(4r) in max-norm of a rank-
r matrix has rank at most
2r−1. We construct
d(d+1) uninformative losses, showing that the cardinality bound is tight up to constants in the large-diameter-to-gap regime, and give a self-contained information-ratio-to-regret transfer that is uniform over fixed measurable selections. Whether the
d5/2 dependence of the monotone case can be retained remains open.