The infinite-family lower-bound conjecture for Spencer discrepancy

For a square $\pm1$ matrix $A$, let

$$\operatorname{disc}(A)=\inf_{x\in\\{\pm1\\}^n}\\|Ax\\|_\infty.$$

Spencer discrepancy lower-bound conjecture. The infinite-family normalized discrepancy satisfies

$$\limsup_{n\to\infty}\sup_{A\in\\{\pm1\\}^{n\times n}}\frac{\operatorname{disc}(A)}{\sqrt n}>1.$$

No infinite family proving a bound strictly larger than $1$ was known in the source, so the question remains open.