Natural communication problems that are hard only for number-in-hand protocols?

I am looking for tools to lower bound the deterministic, blackboard, number-in-hand communication complexity of a certain function which is roughly speaking $$f : \{0,1\}^k \times \ldots \times \{0,1\}^k \to \{0,1\}$$, where $$f$$ has $$k$$ arguments each known by one player in the protocol. I have proven that both the nondeterministic and co-nondeterministic communication complexity of $$f$$ in the number-in-hand model is small, $$\widetilde O(k)$$. But, I conjecture that the deterministic complexity should be as high as possible, $$\Omega(k^2)$$.

Initially, I had hoped that one could resolve the conjecture by embedding a hard two-player communication problem in this model. But, while I haven't finished the proof, I think now it may be the case that for any partition of the $$k$$ inputs of $$f$$ between Alice and Bob, the two-player communication complexity is $$\widetilde O(k)$$ (this was recently shown true for some similar functions).

So, my question is: are there any simple, natural functions $$g : \{0,1\}^k \times \ldots \times \{0,1\}^k \to \{0,1\}$$ for which these types of complexity results hold? Namely, any two-player version of the problem is easy (i.e., $$O(k)$$ complexity), but the $$k$$-player number-in-hand version is known to be hard (i.e., $$\Omega(k^2)$$ complexity)?

(One popular number-in-hand problem is Set-Disjointness, but this does not seem suitable because strong lower bounds require each player to hold many more bits of input, and because the co-nondeterministic complexity of Set-Disjointness is too easy for the problem to embed in the $$f$$ I'm interested in. Earlier, my interest in this same $$f$$ spurred this question, which I would also still appreciate answers to, since the functions brought up in that thread are still somewhat unwieldy. However, if my current suspicion about the two-player protocols for $$f$$ is correct, then the thing that my prior question asks for will be too strong to possibly work: I'd still need something only hard specifically for $$k$$-player number-in-hand where each player holds $$k$$ bits.)