On repdigits as product of $k$-Fibonacci and $k$-Lucas numbers

For an integer $k\geq2$, let $(F_n^{(k)})_{n\geq-(k-2)}$, $(L_n^{(k)})_{n \geq-(k-2)}$ be $k$-Fibonacci and $k$-Lucas sequences, respectively. For these sequences the first $k$ terms are $0,\ldots,0,1$ and $0,\ldots,0,2,1$, respectively, and each term afterwards is the sum of the preceding $k$ terms. In this paper, we determine all possibilities such that $F_n^{(k)} L_m^{(k)}$ can represent a repdigit.