Summary: In a previous paper, the author made a conjecture on the minimal degree of the polynomials, which are identities for the matrix algebra of order \(2n\) with symplectic involution considered as polynomials both in symmetric and skew-symmetric variables, due to the involution. In this paper, the author establishes that the conjecture is not true at least for the case of the matrix algebra of order four by giving an example of such an identity of degree seven which is a Bergman type identity. For the matrix algebra of order six with symplectic involution, the author describes the class of all Bergman type identities both in symmetric and skew-symmetric variables of minimal degree (which happens to be 14).