Let \(T=(T_{1},\dots,T_{n})\) be an \(n\)-tuple of commuting operators on a Hilbert space \(\mathcal H\). For a subset \(E\subset \mathcal H\) denote by \([E]_{T}\) the closed span of the set of vectors \(T_{1}^{k_{1}}\cdots T_{n}^{k_{n}}x\), \(k_{i}\geq 0\), \(i=1,\dots, n\), \(x\in E\). The rank of \(T\) is defined by the formula \( \operatorname{rank}(T)=\min\{ \#E: E\subset {\mathcal H},\, [E]_{T}={\mathcal H} \} \). Let \( {\mathbb D}^{n} = \{ z=(z_{1},\dots z_{n})\in {\mathbb C}^{n}:|z_{i}|=1,\,i=1,\dots, n \} \) be the unit polydisk. Denote by \(M_{z_{i}}\) the operator of multiplication by the coordinate function \(z_{i}\) on the Hardy space \(H^{2}({\mathbb D}^{n})\). A subspace \(S\subset H^{2}({\mathbb D}^{n})\) is said to be invariant if \(M_{z_{i}}S\subset S\) for all \(i=1,\dots, n\). The rank of such a space is defined by \( \operatorname{rank} S = \operatorname{rank}(M_{z_{1}}|_{S},\dots,M_{z_{n}}|_{S}) \). The main result of the paper reads as follows. Theorem. Let \(\varphi, \psi\in H^{\infty}({\mathbb D})\) be the inner functions \(Q_{\varphi} = H^{2}({\mathbb D})\ominus \varphi H^{2}({\mathbb D})\), \(Q_{\psi} = H^{2}({\mathbb D})\ominus \psi H^{2}({\mathbb D})\). Then \(\operatorname{rank}(Q_{\varphi}\otimes Q_{\psi})^{\perp}=2\). This result permits to give a positive answer to the question posed in [R. G. Douglas and R.-W. Yang, Integral Equations Oper. Theory 38, No. 2, 207–221 (2000; Zbl 0970.47016)].