Bruhat-order monotonicity conjecture for the function $f_n$

Let $\mathfrak A_n$ denote the alternating group, let $f_n$ be the function defined by the paper's Algorithm 1, and let $\leq$ denote the Bruhat order on $\mathfrak A_n$. Bruhat-order conjecture. For all $n$ and all $w\in\mathfrak A_n$, one has

$$w\leq f_n(w)$$

in the Bruhat order.

The conjecture extends the verified result for $n\leq 13$; the inversion-number inequality $\operatorname{inv}(w)<\operatorname{inv}(f_n(w))$ is known for every nonidentity $w$, but the full Bruhat-order assertion remains open.