Base-change criterion for fundamental group schemes under purely inseparable extensions

Let $K/k$ be a purely inseparable extension of degree $p$, let $X$ be a connected scheme proper over $k$, and let $x_K\in X_K(K)$ lie over $x\in X(k)$. Let $\mathcal{C}_X$ and $\mathcal{C}_{X_K}$ be the Tannakian categories over $X$ and $X_K$, respectively, and let

$$p_K:X_K\rightarrow X$$

be the natural projection. Suppose that for any $E\in\mathcal{C}_X$, one has $E\otimes_k K\in\mathcal{C}_{X_K}$. Base-change criterion for fundamental group schemes. The following conditions are equivalent:

1. For any $\mathcal{E}\in\mathcal{C}_{X_K}$, $p_*\mathcal{E}\in\mathcal{C}_X$.
2. The natural homomorphism

$$\pi(\mathcal{C}_{X_K},x_K)\rightarrow \pi(\mathcal{C}_X,x)_K$$

is an isomorphism.

This gives a criterion for base change of the fundamental group scheme under a purely inseparable extension of degree $p$. The paper indicates that base change under finite purely inseparable extensions would follow by treating extensions of degree $p$, while the equivalence itself is proposed rather than established.