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 $P_1^{(k)}=1$. For $k\ge 2$ we determine all pairs $(m,n)$ with $n>m\ge 0$ such that $P_n^{(k)}$ and $P_m^{(k)}$ are multiplicatively dependent. The main result states that the only solutions occur for very small $k,m,n$ (which are listed explicitly). The proof uses lower bounds for linear forms in logarithms (Matveev), the Baker-Davenport reduction algorithm, and a computational search.