Sending a lossy version of the innovations process is suboptimal in QG rate-distortion

In the critical range O < D les D_c, the MSE rate-distortion function of a time-discrete stationary autoregressive Gaussian source is equal to that of a related time-discrete i.i.d. Gaussian source. This suggests that perhaps an optimum encoder should compute the related memoryless sequence from the given source sequence with memory and then use a code of rate R(D) to convey the memoryless sequence to the decoder with an MSE of D. In this scenario, the question is, "for D les D_c can a D-admissible code for the original source be obtained via the R-D coding of the innovations process and additional post-processing at the decoder without having to provide any additional information of positive rate?" We show that the answer of this question often is "No"