Projective restriction conjecture for the endomorphism algebra $\mathscr{X}_{m,n}$

Let $\mathscr{X}_{m,n}$ be the endomorphism algebra of the mixed tensor product considered in the paper, let $K(\lambda)$ be a projective module indexed by a bipartition $\lambda$, and let $\operatorname{res}^{m,n}_{m,n-1}$ be the restriction functor induced by the embedding $\mathsf{qw}\kern-1.8pt\mathscr{B}_{m,n-1}\to\mathsf{qw}\kern-1.8pt\mathscr{B}_{m,n}$. Projective restriction conjecture. Restriction of a projective module $K(\lambda)$ over the algebra $\mathscr{X}_{m,n}$ is a sum of projective modules. The statement concerns preservation of projectivity under restriction; the supplied text gives no resolution.