Multiplicative independence in the sequence of $k$-generalized Pell numbers

We study multiplicative dependence between terms of the \(k\)-generalized Pell sequence \((P_n^{(k)})_{n\ge 2-k}\), defined by the linear recurrence \[ P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}, \] with initial conditions \(P_0^{(k)}=\cdots=P_{-(k-2)}^{(k)}=0\) and \(P_1^{(k)}=1\).