The contracted ideal monomiality conjecture

Let $N$ be a positive integer, and let $\mathfrak{J}(N)=J(N)\cap k[x,y]$ be the contracted ideal. The contracted ideal monomiality conjecture.

$$\mathfrak{J}(N)=\langle x^{2N-1},y^{2N-1},x^{2i+1}y^{2j+1}\mid i+j=N-2\rangle.$$

The preceding theorem proves containment of the displayed monomial ideal in $\mathfrak{J}(N)$; the conjecture asserts equality, so it would give an explicit description of the contraction.