Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

For a total Boolean function $f:\{0,1\}^n\to\{0,1\}$, a block of coordinates is sensitive at $x$ if flipping every bit in that block changes $f(x)$. Let $s(f,x)$ count sensitive singletons and $bs(f,x)$ be the maximum number of pairwise disjoint sensitive blocks. Define $s(f)=\max_x s(f,x)$ and $bs(f)=\max_x bs(f,x)$.

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$, indexed by $k=1,2,\ldots$, satisfying

$$s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.$$

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$. Higher is better. The classical exponent is $2$, and Huang's bound gives a ceiling of $4$.

OpenAI's construction achieves an exponent strictly greater than $2$.

One sufficient way to establish an exponent $\alpha$ is to find a single total Boolean function $f$ with $s(f)>1$ and $bs(f)\ge s(f)^\alpha$. Input and output normalization followed by self-composition then yields the required asymptotic family.