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In the Double Pulsar, the Lense-Thirring periastron precession$\dot\omega^\mathrm{LT}$ should be measured by the end of this decade. Theanalyses recently appeared in the literature rely upon a formula for$\dot\omega^\mathrm{LT,\,A}$ induced by the spin angular momentum${\boldsymbol{S}}^\mathrm{A}$ of A valid if the orbital angular momentum$\boldsymbol{L}$ and ${\boldsymbol{S}}^\mathrm{A}$ are aligned, and neglecting$\dot\omega^\mathrm{LT,\,B}$ because of the smallness of${\boldsymbol{S}}^\mathrm{B}$. The impact on $\dot\omega^\mathrm{LT,\,A}$ ofthe departures of the ${\boldsymbol{S}}^\mathrm{A}$-$\boldsymbol{L}$ geometryfrom the ideal alignment is calculated. With the current upper bound on themisalignment angle $\delta_\mathrm{A}$ between them, the angles$\lambda_\mathrm{A},\,\eta_\mathrm{A}$ of ${\boldsymbol{S}}^\mathrm{A}$ areconstrained within $85^\circ \lesssim \lambda_\mathrm{A}\lesssim 92^\circ,\,266^\circ \lesssim \eta_\mathrm{A} \lesssim 274^\circ$. In units of$\dot\omega^\mathrm{GR}=16.89^\circ\,\mathrm{yr}^{-1}$, a range variation aslittle as$\Delta\dot\omega^\mathrm{LT,\,A}\doteq\dot\omega^\mathrm{LT,\,A}_\mathrm{max}- \dot\omega^\mathrm{LT,\,A}_\mathrm{min} = 8\times10^{-8}\,\omega^\mathrm{GR}$ is implied. The experimental uncertainty$\sigma_{\dot\omega_\mathrm{obs}}$ in measuring the periastron rate with theSKA 1-mid telescope should become smaller by 2028-2030. Then, the spatialorientation of ${\boldsymbol{S}}^\mathrm{B}$ is constrained from the existingbounds on the misalignment angle $\delta_\mathrm{B}$, and$\dot\omega^\mathrm{LT,\,B}\simeq 2\times 10^{-7}\,\dot\omega^\mathrm{GR}$ iscorrespondingly calculated. The error $\sigma_{\dot\omega_\mathrm{obs}}$ shouldbecome smaller at the transition from MeerKAT+ facility to SKA 1-mid around2025. (Abridged)

The impact of the spin-orbit misalignment and of the spin of B on the Lense-Thrirring orbital precessions of the Double Pulsar PSR J0737-3039A/B