Optimality of the extended coincidence-based test with nonzero doubleton weight

Let $S_n^{*+}$ be the test statistic defined by adding weighted counts of symbols appearing more than once to the coincidence-based statistic, with finite cutoff $\bar{l}$. The corresponding test is $$

$$\phi^{{*+}}({\text{\protect\boldmath$Z$}}_1)=\mathbb{I}\{S_n^{*+}-{\sf E}_{p}[S_n^{*+}] \geq \tau_n\}.$$

Extended-test optimality conjecture. If $S_n^{*+}$ satisfies $\bar{l}<\infty$, $v_2>-2$, and $v_l\geq 0$ for all $3\leq l\leq\bar{l}$, then the test is optimal in terms of the generalized error exponent. The preceding theorem proves the analogous result when $v_2=0$ and the weights for $l\geq3$ are nonnegative; the conjecture concerns the case $v_2\neq0$ and would establish optimality for a broader family of coincidence-based tests.