The odd-dimensional Sylvester Hadamard discrepancy conjecture

Let $H_k$ be the $2^k\times2^k$ Hadamard matrix obtained by the Sylvester construction, so $H_0=1$ and $H_k=H_1\otimes H_{k-1}$. For $x\in\\{\pm1\\}^{2^k}$, define $\operatorname{disc}(H_k)=\inf_x\\|H_kx\\|_\infty$. Sylvester Hadamard discrepancy conjecture. For odd $k$,

$$\operatorname{disc}(H_k)=\sqrt2\sqrt{2^k}.$$

The even-$k$ case is settled by the construction in the source, while the odd case remains open. If true, this would imply the preceding infinite-family lower-bound conjecture.