This work addresses the issue of Bayesian robustness in the multivariate normal model when the prior covariance matrix is not completely specified, but rather is described in terms of positive semi-definite bounds. This occurs in situations where, for example, the only prior information available is the bound on the diagonal of the covariance matrix derived from some physical constraints, and that the covariance matrix is positive semi-definite, but otherwise arbitrary. Under the conditional Gamma-minimax principle, previous work by DasGupta and Studden shows that an analytically exact solution is readily available for a special case where the bound difference is a scaled identity. The goal in this work is to consider this problem for general positive definite matrices. The contribution in this paper is a theoretical study of the geometry of the minimax problem. Extension of previous results to a more general case is shown and a practical algorithm that relies on semi-definite programming and the convexity of the minimax formulation is derived. Although the algorithm is numerically exact for up to the bivariate case, its exactness for other cases remains open. Numerical studies demonstrate the accuracy of the proposed algorithm and the robustness of the minimax solution relative to standard and recently proposed methods.