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Weaker composition conjecture sufficient to break the 3 log n barrier

Prove that there exist two non-constant Boolean functions f,g: {0,1}^n -> {0,1} such that the depth complexity of their standard block composition satisfies D(f ◇ g) ≥ (1+ε)·n for some constant ε in (0,1).

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Background

To sidestep the difficulty of proving the full KRW conjecture, the paper poses a weaker conjecture that would already imply super-3·log n lower bounds via composition. Establishing a linear-in-n depth lower bound for a single composition f ◇ g would break the 3·log n barrier.

The authors emphasize that this conjecture also remains out of reach even though it is substantially weaker than the full KRW conjecture.

References

Note that we don't even know how to prove a super-$3\log n$ depth lower bound, maybe we should consider following weaker conjecture which suffices to break the $3\log n$ barrier in the first place. \begin{conj}\label{conj:1.2}There exist two non-constant Boolean functions $f,g:{0,1}{n} \rightarrow{0,1}$ such that $\mathsf{D}(f \diamond g) \geq {(1+\epsilon)n}$ for some small constant $\epsilon \in (0,1)$. \end{conj} Unfortunately, we don't even know how to prove this weaker conjecture.

— A nearly-$4\log n$ depth lower bound for formulas with restriction on top (2404.15613 - Wu, 24 Apr 2024) in Conjecture 1.2, Section 1 (Introduction)