The partial-decoupling conjecture for critical points of the DPP likelihood

Let $[N]=\{1,\ldots,N\}$, let $\mathcal S_{[N]}^{++}$ denote the positive-definite symmetric matrices indexed by $[N]$, and let $L^*\in\mathcal S_{[N]}^{++}$. Define

$$K^*=L^*(I+L^*)^{-1}$$

and let $Z\sim\operatorname{DPP}(L^*)$. Let $\Phi_{L^*}$ be the expected log-likelihood function, and call a kernel a partial decoupling of $Z$ when it is obtained by partially decoupling the DPP.

Partial-decoupling conjecture. The kernels of the partial decouplings of $Z$ are the only critical points of $\Phi_{L^*}$.

The preceding theorem establishes that every partial decoupling is a critical point and that strict partial decouplings are saddle points. The conjecture would clarify the global critical-point structure of this non-concave likelihood and could support efficient maximization by first- and second-order methods.