Quantitative Ryser conjecture for circulant matrices

Let $n>4$, let $A$ be an $n\times n$ circulant matrix with entries in $\{-1,1\}$, and let $\|\cdot\|$ denote the Euclidean norm on $\mathbb R^n$. Quantitative Ryser conjecture. There exists $\varepsilon_0>0$ such that there is a vector $x\in\mathbb R^n$ satisfying

$$\left|\|Ax\|-\sqrt n\,\|x\|\right|\geq\varepsilon_0 n^{1/4}.$$

This strengthens Ryser's nonexistence conjecture by asserting a quantitative failure of the Hadamard norm identity. The source notes that the exponent $1/4$ would be optimal up to logarithmic factors if the conjecture holds; the statement is presented as open.