The fixed-point conjecture for the bijection $f_{4,n}\circ f_{1,n}^{-1}$

Let $[k]=\{1,\ldots,k\}$, let $S_n$ be the symmetric group, and let $f_{1,n}$ and $f_{4,n}$ be the bijections from $[1]\times[2]\times\cdots\times[n]$ to $S_n$ defined in the paper. A fixed point of $f_{4,n}\circ f_{1,n}^{-1}$ is a permutation mapped to itself by this composition.

Fixed-point conjecture. There is only $1$ fixed point of $f_{4,n}\circ f_{1,n}^{-1}:S_n\to S_n$, namely the identity permutation, corresponding to the word $1...1$.

This conjecture concerns the fixed-point structure of one of the six compositions of the paper's bijections between words and permutations. The supplied text gives no resolution status.