Charikar–Liu–Liu–Vuong conjecture on balanced grid partitions

Let the $m \times n$ grid graph be partitioned into $k$ connected pieces, and consider the spanning tree distribution on partitions, in which a partition has weight proportional to the product of the numbers of spanning trees in its partition classes. A partition is balanced when all $k$ classes have equal size. Charikar–Liu–Liu–Vuong conjecture. For the $m \times n$ grid graph, the proportion of balanced $k$-partitions under the spanning tree distribution is at least

$$\frac{1}{\operatorname{poly}(m,n)},$$

when $k=O(1)$. If true, rejection sampling from the unrestricted spanning tree distribution would yield an efficient sampler for the balanced distribution; the source presents this as a conjecture motivating its polynomial-time sampling results, and no resolution is supplied here.