Depth-3 circuits for inner product

What is the ${\Sigma }_3^2$-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential $2^{{\alpha }_{n}n}$ for some ${\alpha }_n=\Omega \left(1\right)$. We show that the limiting constant $\alpha ≔\lim \sup {\alpha }_n$ satisfies$0.847...\le \alpha \le 0.965....$ Determining α is one of the seemingly-simplest open problems about depth-3 circuits. The question was recently raised by Golovnev, Kulikov, and Williams (ITCS 2021) and Frankl, Gryaznov, and Talebanfard (ITCS 2022), who observed that $\alpha \in [0.5,1]$. To obtain our improved bounds, we analyse a covering LP that captures the ${\Sigma }_3^2$-complexity up to polynomial factors. In particular, our lower bound is proved by constructing a feasible solution to the dual LP.