Singular value inequalities for generalized anticommutators

We shown among other inequalities that if $A_{1}$ , $B_{1}$ , $X_{1}$ , and $Y_{1}$ are $n\times n$ complex matrices such that $A_{1}$ and $B_{1}$ are positive semidefinite, then $$ s_{j}( Y_{1}A_{1}X_{1}-X_{1}B_{1}Y_{1}) \leq s_{j}(Z \oplus P) ~ \text{for}~j=1,2,\ldots,2n, $$ where $$\begin{aligned}& Z=Z_{1} +|Z_{2}|,~~P=P_{1}+|P_{2}|,\\& Z_{1}=\frac{1}{2} A_{1}^{\frac{1}{2}} (|Y_{1}|^{2}+|X_{1}^{*}|^{2})A_{1}^{ \frac{1}{2}},\\& Z_{2}=\frac{1}{2} B_{1}^{\frac{1}{2}} (X_{1}^{*}Y_{1}-Y_{1}X_{1}^{*})A_{1}^{ \frac{1}{2}},\\& P_{1}=\frac{1}{2} B_{1}^{\frac{1}{2}} (|X_{1}|^{2}+|Y_{1}^{*}|^{2})B_{1}^{ \frac{1}{2}},~ \text{and}\\& P_{2}=\frac{1}{2} A_{1}^{\frac{1}{2}} (Y_{1}^{*}X_{1}-X_{1}Y_{1}^{*})B_{1}^{ \frac{1}{2}}. \end{aligned}$$ This inequality generalizes a recent inequality due to Audeh.