Montgomery-Smith–Semenov hypothesis on signed sums in the integral-uniform norm

Let $f_1,\ldots,f_n\in L_1[0,1]$ satisfy $\|f_i\|_1=1$ for every $i$. For a measurable set $\Delta\subset[0,1]$ with $\mu(\Delta)=2^{-k}$, define the integral-uniform norm by

$$\|g\|'_{2^k}=\sup_{\substack{\Delta\subset[0,1]\mu(\Delta)=2^{-k}}}2^k\int_\Delta |g(x)|\,d\mu(x).$$

Montgomery-Smith–Semenov hypothesis. There exist signs $\theta_i\in\{-1,1\}$ and a constant $c_0>0$ such that

$$\left\|\sum_{i=1}^n\theta_i f_i\right\|'_{2^k}\ge c_0\sqrt{nk}$$

for all $k=1,\ldots,n$. This hypothesis concerns lower bounds for signed combinations of normalized $L_1$ functions; the paper proves it only in a restricted range of $k$ and states that the general form remains open. Random-sign arguments generally do not establish the claimed order for $k\asymp n^\sigma$.