Unimodality conjecture for syzygy statistics of products of projective lines

Let $\b$ be any pair of integers, and let $\d=(d_1,d_2)$. For fixed $k$ and $q$, consider the standard graded Betti numbers, the number of Schur functors counted with multiplicity, and the largest Schur-functor multiplicity in the indicated rows.

Unimodality conjecture. If either $d_1$ or $d_2$ is sufficiently large, then each of the following functions is unimodal:

1. $i\to\beta_{i,i+k}(\P^1\times\P^1,\b;\d)$ for fixed $k$;
2. $p\to$ the total number of Schur functors, counted with multiplicity, appearing in $K_{p,q}(\P^1\times\P^1,\b;\d)$, for fixed $q$;
3. $p\to$ the largest multiplicity of a Schur functor appearing in $K_{p,q}(\P^1\times\P^1,\b;\d)$, for fixed $q$.

These claims are suggested by computed syzygy data; the source does not provide a resolution or a proof.