Hexagonal conjecture for Pleijel's constant

Let $\Omega$ be a bounded planar domain of area $A(\Omega)$, let $(\phi_n)$ be a sequence of Laplace eigenfunctions on $\Omega$, let $\mu(\phi_n)$ denote the number of nodal domains of $\phi_n$, and let $Hexa_1$ be the regular hexagon of area $1$. Define

$$\nu_{Hex}=\frac{4\pi}{\lambda(Hexa_1)}\sim 0.677.$$

Hexagonal conjecture for Pleijel.

$$A(\Omega)\limsup_{n\rightarrow +\infty}\frac{\mu(\phi_n)}{n}\leq \nu_{Hex}.$$

This is proposed as the improved asymptotic bound on the nodal-domain ratio suggested by the hexagonal minimal-partition conjecture; it remains open.