Polterovich's conjecture on the nodal-domain ratio

Let $(\phi_n)$ be a sequence of Laplace eigenfunctions, and let $\mu(\phi_n)$ denote the number of nodal domains of $\phi_n$. Let $Sq_1$ be the square of area $1$, with first eigenvalue $\lambda(Sq_1)$. Polterovich's conjecture.

$$\limsup_{n\rightarrow +\infty}\frac{\mu(\phi_n)}{n}\leq \frac{4\pi}{\lambda(Sq_1)}=\frac{2}{\pi}.$$

The source presents this as a consequence that would follow if suitable bipartite minimal partitions satisfied the stated square-tiling lower bound; it is not established here.