Local $L^{\infty}$ Bernstein conjecture for Laplace eigenfunctions

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$, and let $\varphi_{\lambda}\in C^{\infty}(M)$ satisfy

$$-\Delta_g\varphi_{\lambda}=\lambda\varphi_{\lambda}.$$

For sufficiently small $r>0$, write $B_g(x,r)$ for the geodesic ball centered at $x\in M$. Local $L^{\infty}$ Bernstein conjecture. There is a constant $C>0$ such that

$$\sup_{B_g(x,r)}|\nabla\varphi_{\lambda}|\leqslant C\frac{\sqrt{\lambda}}{r}\sup_{B_g(x,r)}|\varphi_{\lambda}|.$$

More precisely, this is the asserted form of the previously established estimate with the exponent $(d+2)/4$ replaced by $1/2$. The conjecture seeks the sharp dependence on the eigenvalue in a local gradient estimate; the supplied text gives no resolution status.