Idea

Since $\partial_t t=s^{-1}t=s^{-1}s^2\partial_s=s\partial_s$, the

$$\mathcal{H}''_k:=\sum_{j=0}^k(\partial_t t)^j\mathcal{H}''=\sum_{j=0}^k(s\partial_s)^j\mathcal{H}''.$$

are ${\mathbf{C}\{t\}}$-lattices and ${\mathbf{C}\{\!\{s\}\!\}}$-lattices. Since $\mathcal{G}$ is regular, the saturation

$$\mathcal{H}''_\infty\,:=\, \sum_{j=0}^\infty (\partial_t t)^j\mathcal{H}''$$

of $\mathcal{H}''$ is a ${\mathbf{C}\{t\}}$-lattice and, hence, $k_\infty:=\min\{k\ge0\mid\mathcal{H}''_k= \mathcal{H}''_\infty\}$ is a finite number. For any $K\ge n+1$ the inclusions $V^{>-1}\supset\mathcal{H}''\supset V^{n-1}$ imply inclusions

$$V^{>-1}\supset\,\mathcal{H}''_\infty\supset\,\mathcal{H}''\supset... ...n\supset\, \partial_t^{-K}V^{>-1}\supset\, \partial_t^{-K}\mathcal{H}''_\infty$$

and $\mathcal{H}''_\infty$ and $\partial_t^{-K}\mathcal{H}''_\infty$ are $\partial_t t$-invariant. Hence, $\partial_t t$ and $t\partial_t$ induce endomorphisms $\overline{\partial_t t},\, \overline{t\partial_t}\in End_\mathbf{C}\bigl(\mathcal{H}''_\infty\big/ \partial_t^{-K}\mathcal{H}''_\infty\bigr)$ such that the V-filtration $V_{\overline{t\partial_t}}=V_{\overline{\partial_t t}}^{\bullet+1}$ defined by $\overline{t\partial_t}$ on $\mathcal{H}''_\infty\big/\partial_t^{-K}\mathcal{H}''_\infty$ induces the V-filtration on the subquotient $\mathcal{H}''/\partial_t^{-1}\mathcal{H}''=Gr^F_0\mathcal{G}_0$.