Maximal wave front partition conjecture for negative representations

Let $G_{n^*}$ be one of the split classical groups in the source, and let $\sigma_{\neg}$ be a negative representation of $G_{n^*}(F)$. Suppose

$$\operatorname{Jord}(\sigma_{\neg})=\operatorname{Jord}(\sigma_{sn})\cup\\{(\chi_i,n_i),(\chi_i^{-1},n_i):1\leq i\leq t\\}.$$

For $G_{n^*}=\operatorname{Sp}_{2n^*}$ or $\operatorname{O}_{2n^*}$, write

$$\operatorname{Jord}(\sigma_{sn})=\\{(\lambda_0,2n_1+1),\ldots,(\lambda_0,2n_k+1),(1_{\operatorname{GL}_1},2m_1+1),\ldots,(1_{\operatorname{GL}_1},2m_l+1)\\}.$$

For $G_{n^*}=\operatorname{SO}_{2n^*+1}$, write instead

$$\operatorname{Jord}(\sigma_{sn})=\\{(\lambda_0,2n_1),\ldots,(\lambda_0,2n_k),(1_{\operatorname{GL}_1},2m_1),\ldots,(1_{\operatorname{GL}_1},2m_l)\\}.$$

Maximal wave front partition conjecture. If $G_{n^*}=\operatorname{Sp}_{2n^*}$ or $\operatorname{O}_{2n^*}$, then

$$\frak{p}^m(\sigma_{\neg})=\left\{\eta_{\frak{g}_{n^*}^\vee,\frak{g}_{n^*}}\left(\left[(\prod_{j=1}^t n_j^2)(\prod_{i=1}^l(2m_i+1)\prod_{s=1}^k(2n_s+1))\right]\right)\right\}.$$

If $G_{n^*}=\operatorname{SO}_{2n^*+1}$, then

$$\frak{p}^m(\sigma_{\neg})=\left\{\eta_{\frak{g}_{n^*}^\vee,\frak{g}_{n^*}}\left(\left[(\prod_{j=1}^t n_j^2)(\prod_{i=1}^l(2m_i)\prod_{s=1}^k(2n_s))\right]\right)\right\}.$$

Here $\eta_{\frak{g}_{n^*}^\vee,\frak{g}_{n^*}}$ is the duality map between partitions for the dual and original classical Lie algebras, and $[\,\cdot\,]$ denotes the partition notation used in the source. This conjecture predicts the unique maximal wave front partition of a negative representation from its Jordan data; its status is not resolved in the supplied source.