Let \(\mathcal H\) be a separable complex Hilbert space and let \(\mathcal L(\mathcal H)\) denote the set of all bounded operators on \(\mathcal H\). \(T\in \mathcal L(\mathcal H)\) is hyponormal if \(T^*T\geq TT^*\) or, equivalently, if \(\left(\begin{smallmatrix} I&T^*\\T&T^*T\end{smallmatrix}\right)\geq 0\). \(T\) is \(k\)-hyponormal if \((T^{*j}T^i)_{0\leq i,j\leq k}\geq0\). By the Bram-Halmos criterion [\textit{R.\,E.\thinspace Curto, P.\,S.\thinspace Muhly} and \textit{J.--B.\thinspace Xia}, Oper.\ Theory, Adv.\ Appl.\ 35, 1--22 (1988; Zbl 0681.47005)], \(T\) is subnormal (i.e., has a normal extension) iff it is \(k\)-hyponormal for all integers \(k\). In the paper under review, new conditions under which a 2-hyponormal operator is subnormal are given.