Common values of two k-generalized Pell sequences

Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence defined by \begin{equation*} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end{equation*} for $n\geq 2$ with initial conditions \begin{equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,\text{ and }P_{1}^{(k)}=1. \end{equation*} In this study, we look at the equation $P_{n}^{(k)}=P_{m}^{(l)}$ in positive integers $n,m,k,l$ such that $2\leq l<k$ and show that it has only trivial solution, namely $n=m.$