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Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$. Suppose that $K$ is locally acyclic with respect to the composition $g\circ p$, where $p: X\times\mathbb{A}^1\rightarrow X$ is the projection, is $\sigma^* K$ locally acyclic with respect to $g$, where $\sigma$ is the zero section of $p$?

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