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It has been recently proposed a modification of the Skyrme model which admits an exact self-dual sector by the introduction of six scalar fields assembled in a symmetric, positive and invertible 3x3 matrix h. In this paper we study soft manners of breaking the self-duality of that model. The crucial observation is that the self-duality equations impose distinct conditions on the three eigenvalues of h, and on the three fields lying in the orthogonal matrix that diagonalizes h. We keep the self-duality equations for the latter, and break those equations associated to the eigenvalues. We perform the breaking by the addition of kinetic and potential terms for the h-fields, and construct numerical solutions using the gradient flow method to minimize the static energy. It is also shown that the addition of just a potential term proportional to the determinant of h, leads to a model with an exact self-dual sector, and with self-duality equations differing from the original ones by just an additional coupling constant.