Does there exist a $3$-variable word $w$ and $\alpha,\beta,\gamma \in S_n$ with $w(\alpha,\beta,\gamma)=\mathrm{id}$ and three other properties?

Question

Question: For some $n$, does there exist a three-variable word $w(x,y,z)$ over the symmetric group $S_n$ and three non-identity permutations $\alpha,\beta,\gamma \in S_n$ with the following properties? \begin{align*} w(\alpha,\beta,\gamma) &= \mathrm{id} \\ w(\alpha',\beta,\gamma) &= \alpha' & \text{ for all } \alpha' \in S_n \setminus \{\alpha\} \\ w(\alpha,\beta',\gamma) &= \beta' & \text{ for all } \beta' \in S_n \setminus \{\beta\} \\ w(\alpha,\beta,\gamma') &= \gamma' & \text{ for all } \gamma' \in S_n \setminus \{\gamma\} \end{align*}

I'm guessing this is not possible, as it seems we're putting too many constraints on the word.

I'm brainstorming an idea for a secret sharing scheme in which we can identify who submits false shares. For the application I have in mind, $n$ would be around $20$ or so. I'm not sure if the scheme I have in mind will work; there's a lot of matters that would need to be simultaneously resolved (this being one of them).

Answer 1

Probably a stupid answer: As there are no constraints on $\alpha, \beta, \gamma$ you may set $\alpha = \beta = \gamma = id_n \in S_n$ and take the word $w(x,y,z) = xyz$.