Criado–Quintero–Lundqvist–Nenashev dimension conjecture for two additional squares

Let $P_n=cmathbb{F}[x_1,\ldots,x_n]$, let $cell_1,cell_2$ be general linear forms in $P_n$, and let $a_{n,d}$ denote the number of lattice paths from $(0,0)$ to $(n+2-2s,n+2)$, with moves

$$(i,j)\to(i+1,j+1)\quad\text{or}\quad(i,j)\to(i-1,j+1),$$

where the first and last moves are to the right and the path does not cross $x=0$ or $x=n+2-2d$. The Criado–Quintero–Lundqvist–Nenashev conjecture. For all $n,d$, one has

$$\dim\left(\left(\frac{P_n}{(x_1^2,\ldots,x_n^2,\ell_1^2,\ell_2^2)}\right)_d\right)=a_{n,d}.$$

This conjecture gives the dimension predicted for the quotient by $n+2$ general quadratic forms and is connected in the paper to the failure or injectivity of multiplication by a general linear form. The source records it as an existing conjecture and does not supply evidence of resolution.