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Let G be a smooth group scheme over $F_p$ equipped with a $G_m$-action suchthat all weights of $G_m$ on the Lie algebra of G are not greater than 1. Let$Disp_n^G$ be Eike Lau's stack of n-truncated G-displays (this is an algebraicstack over $F_p$). In the case n=1 we introduce an algebraic stack equippedwith a morphism to $Disp_1^G$. We conjecture that if G=GL(d) then the new stackis canonically isomorphic to the reduction modulo p of the stack of 1-truncatedBarsotti-Tate groups of height d and dimension d', where d' depends on theaction of $G_m$ on GL(d).  We also discuss how to define an analog of the new stack for n>1 and how toreplace $F_p$ by $Z/p^n Z$.

On Shimurian generalizations of the stack $BT_1\otimes F_p$