Bufetov's equivalence conjecture for determinantal point process kernels

Let $X$ be the underlying space, let $\mathbb{F}$ be the scalar field, and let $K,Q:X^2\to\mathbb{F}$ be kernels. They are equivalent when, for every $r\geq 1$ and every $(z_1,\dots,z_r)\in X^r$, the cyclic products agree:

$$\prod_{i=1}^r K(z_i,z_{i+1})=\prod_{i=1}^r Q(z_i,z_{i+1}),\qquad z_{r+1}=z_1.$$

Transposition replaces $K(x,y)$ by $K(y,x)$, while conjugation replaces it by $g(x)g(y)^{-1}K(x,y)$ for a function $g:X\to\mathbb{F}\setminus\{0\}$. Bufetov's equivalence conjecture. If $K$ and $Q$ are equivalent kernels, then one can be transformed into the other by a finite sequence of transposition and conjugation transformations.

The conjecture asserts that these canonical transformations account for all equivalent kernels of determinantal point processes. The supplied text gives no evidence of a resolution, so its status remains open.