The small pure resolution conjecture for pairs of partitions

Let $V$ and $W$ be the vector spaces, $R$ the relevant polynomial ring, and let $\lambda^{(0)}\subsetneq\lambda^{(1)}$ be a pair of partitions. Set $c_0=K_{\lambda^{(1)}}(k)$ and $c_1=K_{\lambda^{(0)}}(k)$, so that the two terms have equal rank. A small pure resolution is the equivariant resolution

$$M \leftarrow \mathbb{S}_{\lambda^{(0)}}(V) \otimes \mathbb{S}_{\lambda^{(1)}}(W) \otimes R \leftarrow \mathbb{S}_{\lambda^{(1)}}(V) \otimes \mathbb{S}_{\lambda^{(0)}}(W) \otimes R \leftarrow 0.$$

Small pure resolution conjecture. For any pair of partitions $\lambda^{(0)}\subsetneq\lambda^{(1)}$, such a small pure resolution exists, equivariantly for both $\mathbf{GL}(V)$ and $\mathbf{GL}(W)$.

The conjecture strengthens the realizability theorem by prescribing a particularly short pure resolution. It is known when $|\lambda^{(1)}|=|\lambda^{(0)}|+1$ and when $\lambda^{(1)}$ is obtained from $\lambda^{(0)}$ by adding a connected border strip; the general case is not known.