Steinerberger's Vaserstein distance conjecture for Laplace eigenfunctions

Let $(M,g)$ be a smooth, compact Riemannian manifold without boundary. Let $\phi$ be an $L^2$-normalised eigenfunction of the Laplacian with eigenvalue $L$, satisfying

$$-\Delta \phi=L\phi$$

on $(M,g)$. For $p$ in the range intended by the conjecture, let $W_p$ denote the $p$-Wasserstein distance between the positive and negative parts of $\phi$. Steinerberger's conjecture. Is it true that

$$W_p(\phi^+,\phi^- )\simeq_{p,(M,g)}\frac{1}{\sqrt{L}}\\|\phi\\|_{L^1(M)}^{1/p}?$$

Steinerberger proved an upper bound for $p=1$ with an additional factor of $\sqrt{\log L}$, while this paper obtains the conjectured upper bound for $p=1$ and for high-frequency linear combinations of eigenfunctions. The conjecture for general $p$ remains open in the supplied text.