Sogge–Smith conjecture for nodal extrema of Dirichlet eigenfunctions

Let $(M^n,g)$ be a manifold with boundary, and let $u_\lambda$ be a Dirichlet eigenfunction associated to the eigenvalue parameter $\lambda$. For each nodal domain $A_i$ of $u_\lambda$, let $m_{A_i}$ denote its extremal value, and let $\mathcal{A}(u_\lambda)$ be the collection of nodal domains. If $\alpha(p)$ is the piecewise exponent

$$\alpha(p)=\begin{cases} \left(\frac{2}{3}+\frac{n-2}{2}\right)\left(\frac{1}{4}-\frac{1}{2p}\right),&2\leq p\leq\dfrac{6n+4}{3n-4},\\ \frac{n}{2}\left(\frac{1}{2}-\frac{1}{p}\right)-\frac{1}{4},&\dfrac{6n+4}{3n-4}\leq p\leq+\infty, \end{cases}$$

then Sogge–Smith's conjecture. If $\lambda$ is large enough, there exists $k_g>0$ such that

$$\sum_{i=1}^{|\mathcal{A}(u_\lambda)|}m_{A_i}^p\leq k_g\lambda^{\frac{n}{2}+p\alpha(p)},$$

for any $p\geq2$. This would extend the preceding boundary estimates to the full range $p\geq2$, conditional on the conjectured adapted $L^p$ eigenfunction bounds of Sogge and Smith; the source does not state whether those bounds, or this consequence, have been resolved.