Logarithmic bounds conjecture for maximal directional singular integrals

Let $T_{\mathbf V}$ be the maximal directional singular integral associated with a finite set of $N$ directions ${\mathbf V}$, as defined in the source. Here $p'$ denotes the Hölder conjugate exponent, so $1/p+1/p'=1$, and $C$ and $C_p$ are constants independent of $N$. Logarithmic bounds conjecture. The following bounds hold:

$$\|T_{\mathbf V}\|_{2\to 2,\infty}\lesssim \sqrt{\log N}(\log\log N)^C,$$ $$\|T_{\mathbf V}\|_{p\to p}\lesssim_p(\log N)^{\frac{1}{p'}}(\log\log N)^{C_p},\qquad 2<p<\infty.$$

The authors expect the exponent of $\log N$ to be sharp for every $p>2$; the stated estimates are known for lacunary sets and, in the Hilbert transform case, for Vargas sets, while the general claim remains open.