Rank conjecture for the Fisher information matrix of the Poisson canonical polyadic model

Consider the Fisher information matrix $\mathcal I(\boldsymbol{\theta})$ from the stated theorem, of size $(R\sum_q N_q)\times(R\sum_q N_q)$. Let $\boldsymbol{\theta}$ have strictly positive components, and suppose that no two distinct subsets of its components are linearly dependent.

Fisher information rank conjecture. The rank is

$$\operatorname{rank}(\mathcal I(\boldsymbol{\theta}))=\min\Big(R\sum_q N_q-L,\prod_q N_q\Big),$$

where $L=(\min\{R,N_1,N_2\})^2$ if $P=2$, and $L=R(P-1)$ if $P>2$.

The rank determines the dimensionality of the parameter space relevant to inference and therefore gives insight into the model's complexity and identifiability. The claim is proved in the special case $R=1$, while numerical evidence is provided for ranks greater than one; the general claim remains open in the supplied text.