Conjugation symmetry for stretched Kostka coefficients

Let $\lambda$ and $\mu$ be partitions of the same size. Write $a(\lambda,\mu)$ and $b(\lambda,\mu)$ for the associated invariants introduced in the paper, with $b(\lambda,\mu)$ the degree-related invariant of the stretched Kostka function. Conjugation symmetry conjecture.

$$a(\lambda,\mu)=a(\mu',\lambda'),\qquad b(\lambda,\mu)=b(\mu',\lambda').$$

The claim relates these invariants under conjugation of both partitions and reversal of their order; the supplied text gives no resolution status.