Are there any formulas for the irreducible off-diagonal elements $E^{\lambda}_{ij}$ in the Gelfand-Tsetlin basis of the symmetric group algebra $\mathbb{C}[S_n]$?

Here is the context for my question. There exists well-known formula for *minimal idempotents* (sometimes also called *primitive idempotents*) in the Gelfand-Tsetlin basis (in the sense of Okounkov-Vershik approach) of the symmetric group algebra $\mathbb{C}[S_n]$ originally due to Murphy ["A new construction of Young's seminormal representation of the symmetric group", J. Algebra, 69 (1981), 287-297]:
$$
E_T=\prod_{k=1}^n \prod_{c \neq c_k(T)} \frac{J_k-c}{c_k(T)-c},
$$
where $T$ is a path in the Bratteli diagram of $\mathbb{C}[S_n]$ (or equivalently, *standard Young tableaux* - SYT), $J_k$ is the $k$-th Jucys-Murphy element of $\mathbb{C}[S_n]$, $c_k(T)$ is the content of the box $k$ in SYT $T$, and $c$ runs over all possible contents in all SYTs of size $n$.

If we denote by $\lambda$ the Young diagram of the SYT $T$ and use the index $i$ to refer to the SYT $T$ as a basis vector in the Gelfand-Tsetlin basis of the irreducible representation $\lambda$, then we can think about $E_T$ as being an orthogonal projector $E^{\lambda}_{ii}$ onto diagonal entry $i$ in the block $\lambda$ under Artin-Wedderburn isomorphism $\mathbb{C}[S_n] \cong \bigoplus_{\lambda} \text{End}(V^\lambda)$, where $V^\lambda$ is the irreducible representation labeled by $\lambda$. This naturally raises the question of the existence of a formula for the *off-diagonal* "elementary matrices" $E^{\lambda}_{ij}$ as elements of $\mathbb{C}[S_n]$.