The Strong Sensitivity Conjecture

Let $n$ be a natural number. A Boolean function on $n$ variables is a function $f : \{0,1\}^n \to \{0,1\}$. For $x \in \{0,1\}^n$ and $B \subseteq \{1, \ldots, n\}$, let $\operatorname{flip}(x, B)$ denote the element of $\{0,1\}^n$ obtained from $x$ by flipping all bits whose indices lie in $B$. The local sensitivity of $f$ at $x$, denoted $s(f, x)$, is the number of indices $i \in \{1, \ldots, n\}$ such that $f(\operatorname{flip}(x, \{i\})) \neq f(x)$. The sensitivity of $f$, denoted $s(f)$, is the maximum of $s(f, x)$ over all $x \in \{0,1\}^n$. A collection $\mathcal{B}$ of nonempty subsets of $\{1, \ldots, n\}$ is called a valid block configuration for $f$ at $x$ if the sets in $\mathcal{B}$ are pairwise disjoint and $f(\operatorname{flip}(x, B)) \neq f(x)$ for every $B \in \mathcal{B}$. The local block sensitivity of $f$ at $x$, denoted $\operatorname{bs}(f, x)$, is the maximum cardinality of a valid block configuration for $f$ at $x$. The block sensitivity of $f$, denoted $\operatorname{bs}(f)$, is the maximum of $\operatorname{bs}(f, x)$ over all $x \in \{0,1\}^n$. Does $\operatorname{bs}(f) \leq s(f)^2$ hold for every Boolean function $f$ on $n$ variables?