The fixed-degree matrix dimension conjecture

Fix integers $r\ge1$, $c>2-r$, $t\ge2$, and $d\ge1$. Let $\mathcal{A}$ be a homogeneous $t\times(t+c-1)$ matrix with entries of degree $e$, and set $A=R/I_{t-r+1}(\mathcal{A})$. Suppose $\operatorname{Proj}(A)\in W(\underline{0};\underline{d};r)$, with $\dim A\ge2$ for $c\ne1$ and $\dim A\ge3$ for $c=1$. The fixed-degree matrix dimension conjecture. Then

$$\dim W(\underline{b};\underline{a};r)=t(t+c-1)\binom{e+n}{n}-t^2-(t+c-1)^2+1.$$

This is presented as the fixed-degree specialization of the preceding dimension conjecture; it is motivated by examples and remains open in the stated generality.