Koszul property of diagonal subalgebras

Let S=K[x_1,...,x_n] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f_1,f_2,...,f_k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and Valla [9] concerning Koszul property of the diagonal subalgebras associated to I. Each such subalgebra has the form K[(I^e)_{ed+c}], where c and e are positive integers. For k=3, we extend [9, Corollary 6.10] by proving that K-algebra K[(I^e)_{ed+c}] is Koszul as soon as c >= d/2. We also extend [9, Corollary 6.10] in another direction by replacing the polynomial ring with a Koszul ring.