Conjecture on cocharacter multiplicities with an empty fourth partition

Let $m_{\langle\lambda\rangle}$ denote the multiplicity associated with

$$\langle\lambda\rangle=(\lambda(1),\lambda(2),\lambda(3),\emptyset),$$

where

$$\lambda(1)=(\alpha_1+\alpha_2,\alpha_2),\quad \lambda(2)=(\gamma_1+\gamma_2+\gamma_3,\gamma_2+\gamma_3,\gamma_3)\neq\emptyset,$$

and

$$\lambda(3)=(w_1+w_2,w_2)\neq\emptyset.$$

Cocharacter multiplicity conjecture. The multiplicity $m_{\langle\lambda\rangle}$ is nonzero if and only if either $w_1\leq 2$, or $w_1\geq 3$ and $\gamma_1+\gamma_2\geq\left\lceil\frac{w_1}{2}\right\rceil-1$.

This is intended to characterize the nonzero multiplicities in this family of $*$-cocharacters, building on the preceding example and theorem; the supplied text does not report a resolution.