The dimension conjecture for determinantal loci

Let $t\ge 2$, let $a_1\le\cdots\le a_{t+c-1}$ and $b_1\le\cdots\le b_t$, and let $\mathcal{A}$ be a homogeneous $t\times(t+c-1)$ matrix whose $(i,j)$ entry has degree $a_j-b_i$. Set $A=R/I_{t-r+1}(\mathcal{A})$, where $1\le r\le t-1$ and $2-r\le c$, and suppose $\dim A\ge2$ for $c\ne1$, $\dim A\ge3$ for $c=1$, $\operatorname{Proj}(A)\in W(\underline{b};\underline{a};r)$, $a_1>b_t$, and

$$a_{t+c-1}-b_1<\sum_{i=1}^{t-r+1}(a_i-b_{r+i-1}).$$

The dimension conjecture. Under these hypotheses,

$$\dim W(\underline{b};\underline{a};r)=\lambda_c.$$

This is the paper's principal conjectural dimension formula; the stated numerical inequality is known to force the expected formula in several cases, but the general assertion remains open.