We develop an intrinsic framework for inference on conditional Fréchet means with predictors and responses taking values in finite-dimensional Riemannian manifolds. We derive a second-order expansion of the smoothing bias, represented in the tangent space via the logarithm map, and show that this bias can affect first-order inference. We propose Riemannian robust bias correction (RRBC), which estimates the leading smoothing bias using a local quadratic regression in the response tangent space and corrects the estimator through the exponential map. Our leave-one-out (LOO) covariance estimator uses intrinsic displacements between the full-sample and LOO RRBC estimates, accounting for the variability of the estimated bias correction and its covariance with the local linear estimator. We establish asymptotically valid pointwise confidence sets and simultaneous confidence tubes without undersmoothing. Simulations on spheres and symmetric positive-definite matrices show that RRBC can improve coverage when smoothing bias is nonnegligible. Cardiovascular and ocean-current applications illustrate simultaneous hypothesis testing and inference for smooth functionals.

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