Let H = n 1 , n 2 , n 3 be a numerical semigroup. Let H be the interval completion of H, namely the semigroup generated by the interval n 1 , n 1 +1, . . . , n 3 . Let K be a field and K[H] the semigroup ring generated by H. Let I * H be the defining ideal of the tangent cone of K[H]. In this paper, we describe the defining equations of I * H . From that, we establish the Herzog-Stamate conjecture for monomial space curves stating that β i (I * H ) ≤ β i (I * H ) for all i, where β i (I * H ) and β i (I * H ) are the ith Betti numbers of I * H and I * H respectively.