This paper analyzes the support of the conditional distribution of optimal martingale transport plans in higher dimension. In the context of a distance coupling in dimension larger than 2, previous results established by Ghoussoub, Kim & Lim show that this conditional transport is concentrated on its own Choquet boundary. Moreover, when the target measure is atomic, they prove that the support is concentrated on d+1 points, and conjecture that this result is valid for arbitrary target measure. We provide a structure result of the support of the conditional optimal transport for general Lipschitz couplings. Using tools from algebraic geometry, we provide sufficient conditions for finiteness of this conditional support, together with (optimal) lower bounds on the maximal cardinality for a given coupling function. More results are obtained for specific examples of coupling functions based on distance functions. In particular, we show that the above conjecture of Ghoussoub, Kim & Lim is not valid beyond the context of atomic target distributions.