A counterexample to integer-degree harmonic dimension comparison

For some even n ≥ 8 and integer k ≥ 2, we construct a complete smooth metric on ℝ${}^n$ with nonnegative Ricci curvature whose space of real harmonic functions of pointwise polynomial growth at most k has dimension larger than the Euclidean harmonic-polynomial dimension. The metric is Euclidean near the origin, has asymptotic volume ratio strictly between zero and one, and has nonunique tangent cones at infinity. This answers the integer-degree form of Yau's dimension comparison question in the negative.