Interior $L^p$ estimate conjecture for fractional-Laplacian eigenfunctions

Let $d\geq1$, $0<\alpha<2$, and $2\leq p\leq\infty$. Let $\Omega\subset\mathbb{R}^d$ be a bounded $C^{1,1}$ domain, let $K\subset\subset\Omega$ be compact, and let $e_\lambda$ be an eigenfunction satisfying the paper's eigenvalue equation, with $\lambda>1$. Define

$$\sigma(d,\alpha,p)=\max\left\{\frac{d-1}{2\alpha}\left(\frac12-\frac1p\right),\frac{d-1}{2\alpha}-\frac{d}{\alpha p}\right\}.$$

Interior $L^p$ estimate conjecture. One should have

$$\|e_\lambda\|_{L^p(K)}\lesssim\lambda^{\sigma(d,\alpha,p)}\|e_\lambda\|_2,$$

where the constant is independent of $\lambda$. The theorem in the paper proves this in substantial parameter ranges, but the cases $d=1$, $0<\alpha<\frac14$, $\frac{2(1-2\alpha)}{1-4\alpha}<p\leq\infty$, and $d\geq2$, $0<\alpha\leq\frac12$, $2<p\leq\infty$ remain unresolved in the stated results; at $\alpha=\frac12$ estimates with logarithmic losses are obtained.