Nonexistence of product-type half-flag-transitive automorphism groups

Let $\mathcal D$ be a $2$-$(v,k,\lambda)$ design with replication number $r$ and $\lambda\geq (r,2\lambda)^2$. An automorphism group of $\mathcal D$ is half-flag-transitive when it is transitive on points and on blocks but not on incident point-block flags. It is of product type when it belongs to the product-type case for the induced permutation action on the design. Product-type exclusion conjecture. Any half-flag-transitive automorphism group of $\mathcal D$ is not of product type. The conjecture is motivated by Magma computations for parameter values $v_0\in\{5,7,9,11\}$, which detected no such $2$-design; its general validity remains open.