Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$.
C.D. Sogge proved that we have the following 
$$
\|e_{\lambda}\|_{L^p(M)}\leq C\lambda^{(\frac{1}{2}-\frac{1}{p})}\|e_{\lambda}\|_{L^2(M)}, \quad 2\leq p\leq 6,\\
\|e_{\lambda}\|_{L^p(M)}\leq C\lambda^{2(\frac{1}{2}-\frac{1}{p})-\frac{1}{2}}\|e_{\lambda}\|_{L^2(M)}, \quad 6\leq p\leq \infty.
$$
Such estimates are sharp because for the round sphere $S^2$, the first one is sharp because of the highest weight spherical harmonics and the second one is sharp due to the zonal functions on $S^2$. A natural question is if we can get improved estimates under some additional assumption, indeed, if the manifold is everywhere nonpositive, then again sogge can show that, when $2<p< 6$, the above bound can improve to be $o(\lambda^{(\frac{1}{2}-\frac{1}{p})})$, but the endpoint case $p=6$ is not known,  however, it's valid for the standard torus $\mathbb T^2$ due to a result by Zygmund who showed that one has $\frac{\|e_{\lambda}\|_{L^4(\mathbb T^2)}}{\|e_{\lambda}\|_{L^2(\mathbb T^2)}}=o(1)$ and interpolation with $p=\infty$

My question is that why is it so hard to prove the improved bound for  $p=6$ in general with nonpositive curvature ?