Based on independently distributed $X_1 \sim N_p(θ_1, σ^2_1 I_p)$ and $X_2 \sim N_p(θ_2, σ^2_2 I_p)$, we consider the efficiency of various predictive density estimators for $Y_1 \sim N_p(θ_1, σ^2_Y I_p)$, with the additional information $θ_1 - θ_2 \in A$ and known $σ^2_1, σ^2_2, σ^2_Y$. We provide improvements on benchmark predictive densities such as plug-in, the maximum likelihood, and the minimum risk equivariant predictive densities. Dominance results are obtained for $α-$divergence losses and include Bayesian improvements for reverse Kullback-Leibler loss, and Kullback-Leibler (KL) loss in the univariate case ($p=1$). An ensemble of techniques are exploited, including variance expansion (for KL loss), point estimation duality, and concave inequalities. Representations for Bayesian predictive densities, and in particular for $\hat{q}_{π_{U,A}}$ associated with a uniform prior for $θ=(θ_1, θ_2)$ truncated to $\{θ\in \mathbb{R}^{2p}: θ_1 - θ_2 \in A \}$, are established and are used for the Bayesian dominance findings. Finally and interestingly, these Bayesian predictive densities also relate to skew-normal distributions, as well as new forms of such distributions.