Drinfeld's extension conjecture for Lau's group scheme

Let $d,d'$ be integers with $0\le d'\le d$, let $G=GL(d)$ carry the specified ${\mathbb G}_m$-action, and let $\operatorname{Disp}_1^G$ be the stack of $1$-truncated $G$-displays. Let $\operatorname{Lau}_1^{G,\,\operatorname{true}}$ be the commutative locally free finite group scheme banding Eike Lau's gerbe $\operatorname{BT}_1^{d,d'}\otimes{\mathbb F}_p\to\operatorname{Disp}_1^G$, and let $\operatorname{Lau}_1^G$ be the group scheme constructed in the paper. Their restrictions are canonically isomorphic over the generic locus of $\operatorname{Disp}_1^G$.

Drinfeld's extension conjecture. The isomorphism between these restrictions extends to an isomorphism over the whole $\operatorname{Disp}_1^G$.

The conjecture compares the group scheme arising from Lau's Dieudonné-theoretic description with the paper's proposed group scheme. If true, it implies that the associated gerbes are banded by the same group scheme; the source gives no resolution.