Bigraded Fröberg conjecture for generic forms on $\mathbb P^1\times\mathbb P^1$

Let $R=k[x_0,x_1,y_0,y_1]$ with $\deg x_i=(1,0)$ and $\deg y_i=(0,1)$. Let $f_1,\ldots,f_r$ be generic bigraded forms of bidegrees $(d_i,e_i)$. For a formal power series $A(x,y)=\sum_{i,j\ge0}a_{i,j}x^iy^j$, define $A(x,y)_+=\sum b_{i,j}x^iy^j$, where $b_{i,j}=a_{i,j}$ if $a_{k,l}>0$ for all $(k,l)\le(i,j)$, and $b_{i,j}=0$ otherwise.

Bigraded Fröberg conjecture. The Hilbert series of $R/(f_1,\ldots,f_r)$ is

$$\left(\frac{\prod_{i=1}^r(1-x^{d_i}y^{e_i})}{(1-x)^2(1-y)^2}\right)_+.$$

The source says the analogous guess fails in general, with a counterexample on $\mathbb P^1\times\mathbb P^2$, but conjectures it for $\mathbb P^1\times\mathbb P^1$. Its resolution status is not otherwise specified.