Buchsbaum–Eisenbud conjectural resolution for Schur powers of a determinantal cokernel

Let $R$ be a graded ring, let $F$ and $G$ be graded free $R$-modules of ranks $t$ and $t+c-1$, respectively, and let $\varphi^*:G^*\longrightarrow F^*$ be a graded morphism. Suppose that $\operatorname{depth}_{I_{t-1}(\varphi)}R=2(c+1)$. For the cokernel $\operatorname{coker}(\varphi^*_{p-1,1})$, and in particular for $\bigwedge^2M$, consider the graded free resolution beginning with the displayed complexes.

Buchsbaum–Eisenbud conjectural resolution. A resolution of $\operatorname{coker}(\varphi^*_{p-1,1})$ should always start out as

$$\cdots\longrightarrow L_2^{p-1}(G)^*\oplus A^p(F)\otimes\bigwedge^tG^*\longrightarrow\bigwedge^{p-1}G^*\otimes F^*\longrightarrow\bigwedge^pF^*\longrightarrow\operatorname{coker}(\varphi^*_{p-1,1})\longrightarrow0.$$

In particular, a resolution of $\bigwedge^2M$ should always start out as

$$\cdots\longrightarrow L_2^1(G)^*\oplus A^2(F)\otimes\bigwedge^tG^*\longrightarrow G^*\otimes F^*\longrightarrow\bigwedge^2F^*\longrightarrow\bigwedge^2M\longrightarrow0.$$

Here $L_p^qF=\ker(\delta_{p+1}^{q-1})$, and $A^p(F)=\operatorname{coker}(\bigwedge^{t-p}F\longrightarrow\bigwedge^{t-p+1}F\otimes F^*)$. The claim predicts the initial terms of minimal resolutions, generalizing the known Buchsbaum–Eisenbud descriptions; the source presents it as a prediction rather than a proved result.