Conformally covariant differential operators for the diagonal action of O(p, q) on real quadrics

Let $X=G/P$ be a real projective quadric, where $G=O(p,q)$ and $P$ is a parabolic subgroup of $G$. Let $\left(\pi_{\lambda,\epsilon}, \mathcal{H}_{\lambda,\epsilon}\right)_{ (\lambda,\epsilon)\in \mathbb {C}\times \{\pm\}}$ be the family of (smooth) representations of $G$ induced from the characters of $P$. For $(\lambda, \epsilon), (\mu, \eta)\in \mathbb{C} \times \{\pm\}$, a differential operator $\mathbf{D}_{(\lambda,\epsilon), (\mu,\eta)}^{reg}$ on $X\times X$, acting $G$-covariantly from $\mathcal{H}_{\lambda,\epsilon} \otimes \mathcal{H}_{\mu, \eta}$ into $\mathcal{H}_{\lambda+1,-\epsilon} \otimes \mathcal{H}_{\mu+1, -\eta}$ is constructed.