The planar bichromatic graph algebraic-connectivity conjecture

Let $G$ be a planar bichromatic graph, meaning a planar graph whose vertices can be coloured with two colours so that adjacent vertices have different colours. Let $a(G)$ denote its algebraic connectivity. Bichromatic planar algebraic-connectivity conjecture.

$$a(G)\leq 2.$$

The preceding discussion gives the weaker bound $a(G)\leq 3$; this conjecture proposes that the bound can be improved to two.