Let \(R\) be a semiprimary ring and let \(J\) denote the Jacobson radical. It is shown that if the left generalized projective dimension of \(R/J^ 2\) is finite, then the injectively defined left finitistic dimension of \(R\) is finite. Here a left module is said to have finite generalized projective dimension if there exists an integer \(m\geq 0\) such that \(\text{Ext}^{n + 1} (M,N) =0\) for any left \(R\)-module \(N\) with finite injective dimension.