Optimal logarithmic power conjecture for Lorentz-Sobolev multiplier conditions

Let $1<p<\fty$, let $\Psi$ be as in the main multiplier theorem, let $0<s_1\leq s_2\leq\cdots\leq s_n<1$, and suppose that exactly $d$ of the numbers $s_2,\dots,s_n$ equal $s_1$, with $s_1>|1/p-1/2|$. Let $\phi_{s_1,(1-s_1)d}$ be a concave function satisfying

$$\phi_{s_1,(1-s_1)d}(t)\approx t^{s_1}\log^{(1-s_1)d}\bigl(e+\tfrac1t\bigr),\qquad t>0.$$

For a bounded function $\sigma\in L^\infty(\mathbb R^n)$, define

$$K':=\sup_{j_1,\dots,j_n\in\mathbb Z}\left\|\Gamma(s_1,\dots,s_n)\bigl[\widehat{\Psi}\,D_{j_1,\dots,j_n}\sigma\bigr]\right\|_{\Lambda_{\phi_{s_1,(1-s_1)d}}(\mathbb R^n)}.$$

Optimal logarithmic power conjecture. If $K'<\infty$, then there is a constant $C=C(s_1,\dots,s_n,p,n,d,\psi)$ such that every $f\in\mathscr C_0^\infty(\mathbb R^n)$ satisfies

$$\left\|T_\sigma f\right\|_{L^p(\mathbb R^n)}\leq CK'\left\|f\right\|_{L^p(\mathbb R^n)}.$$

This speculation concerns lowering the logarithmic exponent in the sufficient multiplier condition from $d$ to $(1-s_1)d$. The theorem preceding it proves the corresponding $L^p$ bound with Lorentz function $\phi_{s_1,d}$; whether this sharper exponent always suffices is left open.