Casimir-family criterion for generic Kronecker bihamiltonian structures

Let $M$ be a manifold with compatible Poisson structures $\{,\}_{1}$ and $\{,\}_{2}$. Let $L$ be a finite set with $r$ elements, and let $F_{l,\lambda}$, for $l\in L$ and $\lambda\in\Bbb C$, be smooth functions on $M$ that are Casimirs of $\lambda\{,\}_{1}+\{,\}_{2}$. Suppose

$$F_{l,\lambda}(m)=\sum_{k=0}^{d_l}f_{l,k}(m)\lambda^k.$$

For $m\in M$, let $W_1(m)\subset T_m^*M$ be spanned by the differentials $df_{l,k}|_m$. If, at $m_0\in M$, $\dim W_1(m_0)\geq(\dim M+r)/2$, if some combination $\lambda_1\{,\}_1+\lambda_2\{,\}_2$ has at most $r$ independent Casimir functions near $m_0$, and if $\sum_{l\in L}(2d_l+1)\leq\dim M$, then $\dim M-r$ is even, $\dim W_1(m_0)=(\dim M+r)/2$, $2\sum_{l\in L}d_l+r=\dim M$, and the bihamiltonian structure is Kronecker of type $\left(2d_1+1,\dots,2d_r+1\right)$ on an open subset $U\subset M$ whose closure contains $m_0$. Casimir-family criterion conjecture. Under these hypotheses, all the stated conclusions hold. The criterion would characterize Kronecker structures through the mutual position of Casimir functions for linear combinations of the two Poisson brackets. The source gives no resolution.