3 problems

Halo conjecture. The differentiation basis $\mathcal B$ should differentiate every function $f$ such that

Density-basis differentiation conjecture. If $M_{\mathcal{B}}f(x)<\infty$ almost everywhere, then

Strong differentiation conjecture. If $\widetilde{M}_{\mathcal{B}}f(x)$ is finite almost everywhere, then $f$ is strongly differentiable.