The communication-complexity equivalence conjecture for partition and exactness

For positive integers $n$ and $k$, let $Part_{n,k}$ denote the problem of determining whether pairwise disjoint sets $S_1,\ldots,S_k$ form a partition of $[n]$, and let $Exactly_{n,k}$ denote the corresponding problem of determining whether their union is exactly $[n]$. Write $D_k(f)$ for the deterministic $k$-party number-on-the-forehead communication complexity of a function or problem $f$. Partition–exactness conjecture.

$$D_k(Part_{n,k}) = \Theta(D_k(Exactly_{n,k})).$$

The conjecture expresses the expectation that verifying that pairwise disjoint sets form a partition has communication complexity of the same order as verifying exact coverage. The source presents this as an open question; no resolution is stated.