General recursion formula for derivatives of local representation densities

Let $q$ be the residue-field cardinality, and let $A_{\alpha\beta\gamma\delta}$ denote the relevant diagonal hermitian form with $\alpha>\beta\geq\gamma\geq\delta\geq0$. Let $A'_{0000}(A_{\alpha\beta\gamma\delta})$ be the derivative quantity and let $A_{\lambda\kappa\mu}(A_{\beta\gamma\delta})$ and $A_{1110}(A_{\alpha\lambda\kappa\mu})$ be the local representation densities appearing in the formula. The general local-density recursion conjecture. Under these inequalities,

$$\begin{aligned} A'_{0000}(A_{\alpha\beta\gamma\delta})={}&A_{100}(A_{\beta\gamma\delta})A'_{0000}(A_{\alpha100})\\ &+\sum_{\lambda\geq3}A_{\lambda00}(A_{\beta\gamma\delta})\left\{A'_{0000}(A_{\alpha\lambda00})-A'_{0000}(A_{\alpha(\lambda-2)00})\right\}\\ &+(1-q^2)\sum_{\lambda\geq\kappa\geq1}A_{\lambda\kappa0}(A_{\beta\gamma\delta})A_{1110}(A_{\alpha\lambda\kappa0})\\ &+(1+q)(1-q^2)\sum_{\lambda\geq\kappa\geq\mu\geq1}A_{\lambda\kappa\mu}(A_{\beta\gamma\delta})A_{1110}(A_{\alpha\lambda\kappa\mu}). \end{aligned}$$

The formula is conjectured from the preceding example as a general recursion for local representation densities. Its validity beyond the demonstrated example is not established in the supplied text.