Goulden–Rattan square-coefficient inequalities for expanders

Let $Y_k(u)$ be the set of expanders with $k$ edges, $u-1$ white vertices and one black vertex. Let $X_k(i,j)$ be the set of expanders with $k$ edges, $i+j-2$ white vertices and two black vertices with weights $i-1$ and $j-1$. Goulden–Rattan's square-coefficient conjecture. For natural numbers $i\neq j$ and any natural number $k$,

$$(2j-1)\left\|X_k(j,j)\right\|\geq (j-1)^2\left\|Y_k(2j)\right\|$$

and

$$(i+j-1)\left\|X_k(i,j)\right\|\geq 2(i-1)(j-1)\left\|Y_k(i+j)\right\|.$$

These inequalities reformulate the Goulden–Rattan conjecture for square coefficients in terms of expanders. The supplied text does not state whether these inequalities have been proved or refuted, so their status remains open.