Classical sampling from noisy boson sampling and negative probabilities

It is known that, by accounting for the multiboson interferences up to a finite order, the output distribution of noisy boson sampling, with distinguishability of bosons serving as noise, can be approximately sampled from in a time polynomial in the total number of bosons. The drawback of this approach is that the joint probabilities of completely distinguishable bosons, i.e., those that do not interfere at all, have to be computed also. In trying to restore the ability to sample from the distinguishable bosons with computation of only the single-boson probabilities, one faces the following issue: the quantum probability factors in a convex-sum expression, if truncated to a finite order of multiboson interference, have, on average, a finite amount of negativity in a random interferometer. The truncated distribution does become a proper one, while allowing for sampling from it in a polynomial time, only in a vanishing domain close to the completely distinguishable bosons. Nevertheless, the conclusion that the negativity issue is inherent to all efficient classical approximations to noisy boson sampling may be premature. I outline the direction for an enitre new program, which seems to point to a solution. However, its success depends on the asymptotic behavior of the symmetric group characters, which is not known.