*-polynomial identities of 4x4 upper triangular matrices with the reflection involution

Let $U{T}_4(F)$ be $4\times 4$ upper triangular matrix algebra over a field $F$ of characteristic zero and let $\mathcal{A}$ be the subalgebra of $U{T}_4(F)$ linearly generated by ${$\mathbf{e}${ij}:1 \leq i\leq j \leq 4 } \setminus $\mathbf{e}${23}$where${$\mathbf{e}$_{ij} : 1 \leq i\leq j \leq 4}$ is the standard basis of $U{T}_4(F)$. We describe the set of all $\ast$-polynomial identities for $\mathcal{A}$ with the involution defined by the reflection of second diagonal.