Let $f$ be a random completely multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random. Is it true that$$\underset{N\to \infty }{\mathrm{limsup}}\frac{{\sum }_{m\le N}f(m)}{\sqrt{N}}=\infty$$with probability $1$?