A polynomial criterion for Jordan elements in a free associative algebra

Let $A=Φ\langle X\rangle$ be a free associative algebra over a field of characteristic zero, and let $J$ be the Jordan subalgebra of $A^{(+)}$ generated by $X$ and $1$. For every $n\geq1$ we construct an element $U_n\in\mathbb Q[\mathfrak S_n]$ whose image on $A_n$ is exactly $J_n$. If \[ \det(tI-