Syzygies of secant ideals of Pl\"ucker-embedded Grassmannians are generated in bounded degree

Over a field of characteristic $0$, we prove that for each $r\ge 0$ there exists a constant $C(r)$ so that the prime ideal of the $r$th secant variety of any Pl"ucker-embedded Grassmannian $\mathbf{G}\mathbf{r}(d,n)$ is generated by polynomials of degree at most $C(r)$, where $C(r)$ is independent of $d$ and $n$. This bounded generation ultimately reduces to proving a poset is noetherian, we develop a new method to do this. We then translate the structure we develop to the language of functor categories to prove the $i$th syzygy module of the coordinate ring of the $r$th secant variety of any Pl"ucker-embedded Grassmannian $\mathbf{G}\mathbf{r}(d,n)$ is concentrated in degrees bounded by a constant $C(i,r)$, which is again independent of $d$ and $n$.