Uniform heat-content conjecture for nodal sets

Let $(M,g)$ be a compact smooth manifold without boundary. Let $u$ be a Laplacian eigenfunction with eigenvalue $\lambda$, and let $p_t(x)$ be globally defined with respect to its nodal set, with its restriction to each nodal domain given by the corresponding local function. Uniform heat-content conjecture. There exists a constant $c>0$ depending only on $(M,g)$ such that

$$p_{\lambda^{-1}}(x)>c\qquad\text{for all }x\in M.$$

The authors state that this conjecture seems likely but should be extremely difficult; together with heat content isoperimetry at $t=\lambda^{-1}$, it would imply the lower-bound half of Yau's conjecture.