Refined exponent conjecture for simple shapes

Let $\Delta=(T_i,d_i,\lambda_i,\eta_i)_{1\leq i\leq k}\neq\Sigma_{\lambda,\eta}$ satisfy the setup and conditions of the cited theorems, and let $R_0(\Delta)$ be

$$R_0(\Delta):=\frac12\left(N^2-\sum_iT_i^2d_i^2\right)+\sum_i\left(T_i^2+\frac12T_i(T_i-1)(d_i^2-1)\right)$$ $$=\bar R(\Delta)-\sum_i\left(\frac12T_i^2d_i(d_i+1)-\left(T_i^2+\frac12T_i(T_i-1)(d_i^2-1)\right)\right).$$

Refined exponent conjecture. Under all the same conditions, the bound in Theorem may be tightened to

$$S^{|G|}_\Delta(\operatorname{EP}_\lambda f^\infty_{\mathfrak n})=O(|\mathfrak n|^{R_0(\Delta)}q_{S_1}^{A+B\kappa}).$$

The conjecture is motivated by the expected Gelfand–Kirillov dimensions of Speh representations; the paper identifies uniform bounds on local character coefficients as the main obstacle to proving it.