Normality of monodromy group in generic convolution group

On an abelian variety $A$, sheaf convolution gives a Tannakian formalism for perverse sheaves. Let $X$ be an irreducible algebraic variety with generic point $\eta$. Let $K$ be a family of perverse sheaves (more precisely, a relative perverse sheaf) on the constant abelian scheme $p_X:A\times X\to X$. We show that for uncountably many character sheaves $L_{\chi}$ on $A$, the monodromy groups of $R^0p_{X*}(K\otimes p_A^*L_{\chi})$ are normal in the Tannakian group $G(K|_{A_{\eta}})$ of the perverse sheaf $K|_{A_{\eta}}\in\mathrm{Perv}(A_{\eta})$. This result is inspired from and could be compared to two other normality results: In the same setting, the Tannakian group $G(K|_{A_{\bar{\eta}}})$ is normal in $G(K|_{A_{\eta}})$ (due to Lawrence-Sawin). For a polarizable variation of Hodge structures, outside a meager locus, the connected monodromy group is normal in the derived Mumford-Tate group (due to Andr\'e).