Almost-equal Kronecker blocks conjecture for semi-direct sums of $\operatorname{sl}(n)$

Let $\mathfrak{g} = \operatorname{sl}(n) \ltimes \left(\mathbb{C}^n\right)^k$ be the semi-direct sum given by the standard representation, with $k<n$ and

$$n=kd+r.$$

Assume $1<r<k-1$. Almost-equal Kronecker blocks conjecture. The Jordan--Kronecker invariants of $\mathfrak{g}$ have no Jordan blocks and consist of

$$\operatorname{ind}\mathfrak{g}=kr-r^2+1$$

Kronecker blocks whose sizes differ by at most $1$. More precisely, if

$$\frac{\dim\mathfrak{g}+\operatorname{ind}\mathfrak{g}}{2}=l\operatorname{ind}\mathfrak{g}+b,\qquad 0\leq b<\operatorname{ind}\mathfrak{g},$$

then there are $b$ Kronecker blocks of size $2l+1$ and $\operatorname{ind}\mathfrak{g}-b$ Kronecker blocks of size $2l-1$. The preceding theorem establishes the corresponding result when $r=1$ or $r=k-1$, while the cases $1<r<k-1$ are conjectural; the claim concerns the generic skew-symmetric matrix pencils with fixed rank associated with these Lie algebras.