Tangent Space of Moduli of Log-Smooth Curves

Question

We consider an algebraically closed field $\underline{k}$ and all constructions that we will consider are over this field. It is well known that for each relative nodal curve $\underline{f}: \underline{C} \to \underline{S}$ one can assign a minimal log-structure to $\underline{C}$ and $\underline{S}$ upgrading this curve to a log-smooth curve $f: C \to S$. With this construction, one sees that the moduli stack of minimal log-smooth curves $\mathcal{M}_{g}^{\log}$, considered as a Moduli functor from the category of $\underline{k}$-schemes, is equivalent to the stack of nodal curves $\overline{\mathcal{M}}_g$. Furthermore, the tangent space at a $\underline{k}$-point $\underline{C} \to \underline{k}$ of $\overline{\mathcal{M}}_g$ is known to be given by \begin{align*} T_{\underline{C}/\underline{k}}\overline{\mathcal{M}}_g=Ext^1(\Omega_{\underline{C}/ \underline{k}}, \mathcal{O}_\underline{C})= H^1((\Omega_{\underline{C}/ \underline{k}})^\vee) \end{align*} As both stacks are equivalent, we will have $T_{\underline{C}/\underline{k}}\overline{\mathcal{M}}_g= T_{C/k} \mathcal{M}_g^{\log}$. So I think I am justified to expect that this tangent space can be described in a similar way using only "objects that live in the world of log-geometry", such as $H^1((\Omega^{log}_{C/k})^\vee)$. So is it true that I can equivalently describe \begin{align*} T_{\underline{C}/\underline{k}}\overline{\mathcal{M}}_g= H^1((\Omega^{log}_{C/k})^\vee) \end{align*} If yes, is there any reference, if no is there a similar "logarithmic" description of the tangent space?