Conjecture on the complete support of the *-cocharacter of M_{1,2}(F)

Let $F$ be a field of characteristic zero, and let

$$\chi_n^{*}(M_{1,2}(F))=\sum_{\substack{\langle\lambda\rangle\vdash n,\\ h(\lambda(1))\leq 2,\ h(\lambda(2))\leq 3,\\ h(\lambda(3))\leq 2,\ h(\lambda(4))\leq 2}}m_{\langle\lambda\rangle}\chi_{\langle\lambda\rangle}$$

be the $n$th $*$-cocharacter. Let $\langle\lambda\rangle=(\lambda(1),\lambda(2),\lambda(3),\lambda(4))$, with

$$\lambda(2)=(\gamma_1+\gamma_2+\gamma_3,\gamma_2+\gamma_3,\gamma_3),\quad \lambda(3)=(w_1+w_2,w_2),$$ $$\lambda(4)=(\rho_1+\rho_2,\rho_2),\qquad l=\max\{w_1,\rho_1\}.$$

Cocharacter support conjecture. The multiplicity $m_{\langle\lambda\rangle}$ is nonzero if and only if one of the following holds: (i) $\lambda(3)=\lambda(4)=\emptyset$ and $h(\lambda(1))\leq 1$; (ii) $\lambda(j)\neq\emptyset$ for some $j\in\{3,4\}$ and $|w_1-\rho_1|\leq 2$; or (iii) $\lambda(j)\neq\emptyset$ for some $j\in\{3,4\}$, $|w_1-\rho_1|\geq 3$, and $\lambda(2)\neq\emptyset$ with $\gamma_1+\gamma_2\geq\left\lceil\frac{l}{2}\right\rceil-1$.

This conjecture is presented as a synthesis of the preceding conjectures and the final theorem, giving a proposed complete description of the nonzero multiplicities in the $*$-cocharacters of $M_{1,2}(F)$. The supplied text does not establish whether it is resolved.