  We consider the Metropolis biased card shuffling (also called the
multi-species ASEP on a finite interval or the random Metropolis scan). Its
convergence to stationary was believed to exhibit a total-variation cutoff, and
that was proved a few years ago by Labb\'e and Lacoin. In this paper, we prove
that (for $N$ cards) the cutoff window is in the order of $N^{1/3}$, and the
cutoff profile is given by the GOE Tracy-Widom distribution function. This
confirms a conjecture by Bufetov and Nejjar. Our approach is different from
Labb\'e-Lacoin, by comparing the card shuffling with the multi-species ASEP on
$\mathbb{Z}$, and using Hecke algebra and recent ASEP shift-invariance and
convergence results. Our result can also be viewed as a generalization of the
Oriented Swap Process finishing time convergence of Bufetov-Gorin-Romik, which
is the TASEP version (of our result).
