I'm looking for a proof for this problem on simplex which I think it is true

Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $(S)$ is circumscribed sphere of $\mathcal{A}$ with center $O$ and radius $R$. $P$ is a point inside $\mathcal{A}$. Lines $PA_i$ meets $(S)$ again at $B_i$, $i=\overline{0,n}$. Let $r_A$ and $r_B$ be the radius of inscribed sphere of the simplex $\mathcal{A}=A_0A_1...A_n$ and $\mathcal{B}=B_0B_1...B_n$. Prove that $$R^2\ge n^2r_A\cdot r_B+OP^2.$$