Homogeneity conjecture for special strict partitions

A strict partition $\lambda$ is special if all its non-zero parts are congruent to either $1$ or $2$ modulo $3$. Equivalently, it can be written as

$$\lambda=\nu+3\alpha,$$

where $\nu$ is a $3$-bar core and $\alpha$ is a partition with $l(\alpha)\leqslant l(\nu)$. A partition $\alpha$ is $3$-Carter if, for every $1\leqslant r<s$ and every $1\leqslant c\leqslant\alpha_s$, the $(r,c)$-hook length and the $(s,c)$-hook length of $\alpha$ are divisible by the same powers of $3$. A partition $\lambda$ is homogeneous when its associated spin decomposition number has the homogeneity property considered in the paper.

Homogeneity conjecture for special strict partitions. Suppose $p=3$ and $\lambda$ is a special strict partition, written as $\lambda=\nu+3\alpha$ with $\nu$ a $3$-bar core. Then $\lambda$ is homogeneous if and only if $\alpha$ is a $3$-Carter partition.

This conjecture extends the classification of homogeneous strict partitions to the special case excluded from the paper's main theorem. Its status is not resolved in the supplied source context.