$\text{PGL}$ orbits on products of flag varieties

In this paper, we study the existence of a dense orbit for the diagonal $\text{PGL}(n)$ action on self-products of partial flag varieties. We determine when there exists a dense orbit for flag varieties of the form $F(k_1,\dots ,k_r;n{)}^m$ when $k_i=i$ or when $m=3$ and $k_i=\mathrm{\ell }+i$ for some $\mathrm{\ell }\ge 0$. We also show that certain infinite families of products do not have a dense orbit.