KRW Conjecture: separation of NC^1 and P
There are a few points to consider here:

If we wish to show that $CC(f) = \omega(\log n)$, we should show that it is not possible to cover the domain of $f$ using $\mathrm{poly}(n)$ monochromatic rectangles (rather than $n$ rectangles). Showing it for $n$ rectangles would only establish that $CC(f) &gt; \log n$.

If we wish to prove a lower bound on $D(EQ)$, then we should prove a lower bound on $CC(\mathit{KW}_{EQ})$ rather than on $CC(EQ)$. Therefore, the lower bound on $CC(EQ)$ does not imply a lower bound on $D(EQ)$.

In particular, in order to use the Karchmer-Wigderson approach to prove lower bounds on depth complexity, we need to prove lower bounds on the communication complexity of relations rather than functions. This is considerably trickier. For example, in the context of relations, one needs to change the definition of a monochromatic rectangle. Now, it no longer makes sense to require that &quot;the function is constant over $f$&quot; since a relation may allow multiple valid solutions for every input, so it is not clear what being &quot;constant&quot; means. Instead, one should use the following more complicated definition: a rectangle $R \subseteq X \times Y$is said to be monochromatic with respect to a relation $S \subseteq X \times Y \times Z$ if there exists an output $z\in Z$ that is a valid solution for every input in $R$ (i.e., $(x,y,z) \in S$ for every $(x,y) \in R$).

Alas, it is not possible to use rectangle covers to prove meaningful lower bounds on KW relations. The reason is that every KW relations can be covered by $2n$ rectangles. Those rectangles are constructed as follows: for every index $i \in [n]$ and bit $b \in \{0,1\}$ we have the rectangle $$R_{i,b} = \{ x\in f^{-1}(0) \mid x_i = b \} \times \{ y\in f^{-1}(1) \mid y_i = \neg b \}. $$ It is not hard to see that those rectangles are indeed monochromatic rectangles for any KW relation $\mathit{KW}_f$, and that they cover the whole domain of the KW relation.