Angel–Gorin–Holroyd–Romik–Virág conjecture on random reduced words

Let $w_0=[n,n-1,\ldots,1]\in S_n$ be the longest permutation, let $p$ be its inversion number, and choose a reduced word $(a_1,\ldots,a_p)$ uniformly at random. Form the initial product $s_{a_1}s_{a_2}\cdots s_{a_{\lfloor p/2\rfloor}}$ and consider the locations of the $1$'s in its permutation matrix. Random reduced-word limit-shape conjecture. As $n$ tends to infinity, the probability distribution of these $1$'s approaches the surface measure of the sphere projected to two dimensions. The conjecture is stated as still open in the source.