Double-duality and exact-sequence conjecture for triangular t-modules

Let $K$ be a field of characteristic $p>0$, let $\tau$ denote the Frobenius skew-polynomial variable, and let $C$ be the Carlitz module defined by $C_t=\theta+\tau$. For a triangular $\mathbf{t}$-module $\Upsilon$, define its dual by

$$\Upsilon^{\vee}=\operatorname{Ext}^{1}_{0,\tau}(\Upsilon,C).$$

Here $\operatorname{Ext}^{1}_{\tau}$ denotes the corresponding extension module, and $\mathbb{G}_{a,K}^{s}$ is the $s$-fold power of the additive group over $K$.

Double-duality and exact-sequence conjecture. For every triangular $\mathbf{t}$-module $\Upsilon$,

$${\Upsilon^{\vee}}^{\vee}\cong \Upsilon,$$

and there exists an exact sequence of $\mathbf{t}$-modules

$$0\longrightarrow \Upsilon \longrightarrow \operatorname{Ext}^{1}_{\tau}(\Upsilon^{\vee},C) \longrightarrow \mathbb{G}_{a,K}^{s} \longrightarrow 0,$$

for some natural number $s$ depending on $\Upsilon$.

The conjecture proposes a Cartier--Nishi-type duality for triangular $\mathbf{t}$-modules, together with a precise description of the failure of the unrestricted extension module to coincide with the original module. Its validity is supported by the study of triangular modules of dimension two, but the supplied text does not establish the conjecture.