Budzik's conjecture on superalgebra invariant multiplicities

Let $k$ and $\ell$ be nonnegative integers, let $\lambda$ be a partition, and let $m_\lambda(k,\ell)$ denote the multiplicity associated with the hook Schur decomposition for the superalgebra invariant series. Define $m_\lambda'(k,\ell)$ by the complex integral

$$m_\lambda'(k,\ell)=\frac1{k!\ell!(2\pi i)^{k+\ell}}\oint_{|y_i|=1}\oint_{|x_i|=1+\epsilon} HS_\lambda(Z_0;Z_1)\Delta \frac{dx_1}{x_1}\wedge\cdots\wedge\frac{dy_\ell}{y_\ell},$$

where $\Delta$, $Z_0$, $Z_1$, and $HS_\lambda$ are as defined in the surrounding construction. Budzik's conjecture.

$$m_\lambda'(k,\ell)=m_\lambda(k,\ell)-m_\lambda(k-1,\ell-1).$$

This predicts that the complex-integral multiplicity is the difference between the multiplicities for successive hooks. The surrounding discussion states that the equality had been proved for partitions occurring with nonzero multiplicity in the relevant hook representation range, but presents the displayed identity as Budzik's conjecture in general.