Parity-sensitive asymptotic 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 the refined exponent defined above. Parity-sensitive asymptotic conjecture. If all the $d_i$ have the same parity, then for every $\epsilon>0$,

$$C_{\epsilon,1}|\mathfrak n|^{R_0(\Delta)-\epsilon}\leq S^G_\Delta(\operatorname{EP}_\lambda f^\infty_{\mathfrak n})\leq C_{\epsilon,2}|\mathfrak n|^{R_0(\Delta)+\epsilon},$$

for constants $C_{\epsilon,1},C_{\epsilon,2}$ with $C_{\epsilon,2}=O_\epsilon(q_{S_1}^{A+B\kappa})$. If the $d_i$ have different parities, then

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

This conjecture incorporates the cancellation caused by varying signs when the $d_i$ have different parities. The supplied text gives no resolution status.