Hardness of classically sampling one clean qubit model with constant total variation distance error

The one clean qubit model (or the DQC1 model) is a restricted model of quantum computing where only a single input qubit is pure and all other input qubits are maximally mixed. In spite of the severe restriction, the model can solve several problems (such as calculating Jones polynomials) whose classical efficient solutions are not known. Furthermore, it was shown that if the output probability distribution of the one clean qubit model can be classically efficiently sampled with a constant multiplicative error, then the polynomial hierarchy collapses to the second level. Is it possible to improve the multiplicative error hardness result to a constant total variation distance error one like other sub-universal quantum computing models such as the IQP model, the Boson Sampling model, and the Fourier Sampling model? In this paper, we show that it is indeed possible if we accept a modified version of the average case hardness conjecture. Interestingly, the anti-concentration lemma can be easily shown by using the special property of the one clean qubit model that each output probability is so small that no concentration occurs.