Graded Leibniz algebras induced by some filtrations

Question

Let consider a filtration of Leibniz algebra $L$ as $L_0=L \supseteq L_1 \supseteq \dots L_n \supseteq \dots$, with the property that $L_i L_j \subseteq L_{i+j}$. If we consider the grading $gr L = \bigoplus_{i\geq 0} \frac{L_i}{L_{i+1}} $, is it true to say that $gr L$ is a graded Leibniz algebra? In the case of Leibniz algebras, we have a concept known as kernel of Leibniz algebra defined as $Ker(L)=\langle [x,x] \mid x \in L \rangle$ such that $\frac{L}{Ker(L)}$ becomes a Lie algebra.