Positive-Robin gap collapse in smooth convex three-dimensional domains

We give a candidate disproof of the diameter-normalized positive-Robin fundamental-gap conjecture stated as AIM problem \#021. A convex polytope with square end sections and one transverse central protrusion is made thin in two directions. The first transverse Robin energy has leading term equal to the boundary parameter times the fiber perimeter-to-area ratio. That ratio has two reflection-related minima and a central barrier. An elementary uniform fiber estimate, a localized test function, and a ground-state cutoff imply that the full three-dimensional spectral gap tends to zero, although the diameter tends to two. Explicit estimates give a polyhedral gap below $0.021$ at thickness $10^{-12}$, whereas the interval comparison exceeds $2.9$. A specified smooth strongly convex approximation and a quadratic-form convergence argument transfer this strict violation to a single smooth convex domain.