Bakhtiari, Lattimore and Szepesvári (COLT 2025) proved that Thompson sampling (TS) has Bayesian regret $\tilde O(d^{5/2}\sqrt n)$ for bandit convex optimisation with convex \emph{monotone} ridge losses $f(x)=\ell(\ip{x}θ)$, 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 $O\big((d+1)^4\sqrt{dn}\,\log(e+nd\max\{1,\diam K\})\big)=\tilde O(d^{9/2}\sqrt n)$. 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 $O(d^2)$ 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 $d^{5/2}$ dependence of the monotone case can be retained remains open.

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