Weighted maximal operator conjecture for Walsh-Fourier partial sums in $H_1$

Let $H_1$ be the martingale Hardy space, let $S_nF$ denote the $n$th partial sum of the Walsh-Fourier series of $F$, and let $A_s$ be the finite set of indices defined from the binary decompositions of the integers under consideration. For a sequence of positive numbers $\{n_k:k\geq 0\}$, define

$$S^{\ast,\nabla}F:=\sup_{k\in\mathbb{N}}\frac{|S_{n_k}F|}{A_{|n_k|}}.$$

Weighted maximal operator conjecture. For every $f\in H_1$, the operator $S^{\ast,\nabla}$ is bounded from $H_1$ to $L_1$. Moreover, if $\sup_{k\in\mathbb{N}}|A_{n_k}|=\infty$ and $\{\varphi_n\}$ is nondecreasing with

$$\overline{\lim}_{k\to\infty}\frac{A_{|n_k|}}{\varphi_{|n_k|}}=\infty,$$

then there exists a martingale $f\in H_1$ for which the maximal operator

$$\sup_{k\in\mathbb{N}}\frac{|S_{n_k}F|}{\varphi_{|n_k|}}$$

is not bounded from $H_1$ to $L_1$.

The claim proposes both an $H_1\to L_1$ bound with the weight determined by the binary-combinatorial quantity $A_{|n_k|}$ and its sharpness: substantially smaller nondecreasing weights cannot yield boundedness along sequences with unbounded $A_{n_k}$. The parser supplies no resolution evidence, so the conjecture is recorded as open.