The multiplicity criterion for highest weights in the orthosymplectic cocharacter

Let $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$ be a highest-weight label in the decomposition denoted by $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$, where

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

and

$$\lambda(4)=(\rho_1+\rho_2,\rho_2)\neq\emptyset.$$

Here $m_{\langle\lambda\rangle}$ denotes the multiplicity associated with this label. Multiplicity criterion. One has

$$|w_1-\rho_1|\leq 2$$

if and only if

$$m_{\langle\lambda\rangle}\neq 0.$$

This conjecture is motivated by an example where $|w_1-\rho_1|=4$ and the corresponding multiplicity vanishes; it proposes that the bound $2$ exactly characterizes the nonzero multiplicities in this family. The source does not indicate whether the conjecture has been resolved.