McLeman's Massey-product criterion for finiteness of p-class field towers

Let $p>3$ be a prime, and let $K$ be an imaginary quadratic field whose class group has $p$-rank two. Set

$$G=\operatorname{Gal}(K^{\operatorname{ur},p}/K).$$

Choose a basis $x,y$ of $H^1(G,\mathbb{Z}/p\mathbb{Z})$, and let $\rho_1,\rho_2$ denote the representations whose trace functionals occur below. McLeman's Massey-product criterion. The group $G$ is finite if and only if

$$\begin{pmatrix} \operatorname{tr}_{\rho_1}\langle x,x,y\rangle & \operatorname{tr}_{\rho_1}\langle y,y,x\rangle \\ \operatorname{tr}_{\rho_2}\langle x,x,y\rangle & \operatorname{tr}_{\rho_2}\langle y,y,x\rangle \end{pmatrix}$$

is invertible. The source presents this as an equivalent version of the Zassenhaus type conjecture and states that it will disprove it; hence this criterion is refuted in the stated generality.