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It was shown a few years back that for a stationary regular black hole orstar solution in the Brans-Dicke theory with a positive cosmological constant$\Lambda$, endowed with a de Sitter or cosmological event horizon in theasymptotic region, not only there exists no non-trivial field configurations,but also the inverse Brans-Dicke parameter $\omega^{-1}$ must be vanishing.This essentially reduces the theory to Einstein's General Relativity. Theassumption of the existence of the cosmological horizon was crucial for thisproof. However, since the Brans-Dicke field $\phi$, couples directly to the$\Lambda$-term in the energy-momentum tensor as well as $\Lambda$ acts as asource in $\phi$'s equation of motion, it seems reasonable to ask : can $\phi$become strong instead and screen the effect of $\Lambda$, at very large scales,so that the asymptotic de Sitter structure is replaced by some alternative, yetstill acceptable boundary condition? In this work we analytically argue that nosuch alternative exists, as long as the spacetime is assumed to be free of anynaked curvature singularity. We further support this result by providingexplicit numerical computations. Thus we conclude that in the presence of apositive $\Lambda$, irrespective of whether the asymptotic de Sitter boundarycondition is imposed or not, a regular stationary black hole or even a starsolution in the Brans-Dicke theory always necessitates $\omega^{-1}=0$, andthereby reducing the theory to General Relativity. The qualitative differencesof this result with that of the standard no hair theorems are also pointed out.

Stationary black holes and stars in the Brans-Dicke theory with $\Lambda >0$ revisited