An identity for generalized q-Fibonacci polynomials

Question

Let $k$ be a positive integer and consider the generalized $q-$Fibonacci polynomials $F_n^{(k)}(x)$ which satisfy the recursion $F_n^{(k)}(x) = xF_{n - 1}^{(k)}(x) + q^{n-k} F_{n - k}^{(k)}(x)$ with initial values $F_n^{(k)}(x) = {x^n}$ for $0 \leqslant n < k.$ Denote by $\binom{n}{k}_q$ the $q-$binomial coefficients. I need the identity $$\sum\limits_{j = 0}^{n - 1} {{{( - 1)}^{n - 1 - j}}} {x^k}\binom{n-k}{n-j-1}_q {q^{(k - 1)\left(\binom{n}{2}-\binom{j+1}{2} \right)}}F_{kj}^{(k)}(x) = {x^n}F_{(k - 1)n}^{(k)}(x).$$ Has anyone an idea for a simple proof?