Exactness of Some Completed Tensor Products

Question

Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/1501.02215 it is stated that, for every flat $O(X)^{\circ}$-module $Q$, the functor
$$A\mapsto \widehat{A\otimes_{O(X)^{\circ}}Q}\otimes_{\mathcal{R}}k$$
where the completion is with respect to the $\omega$-adic topology, is left exact. Showing this would involve showing that the kernel of the compled tensor is torsion, however, I do not understand why this is the case.

Answer 1

This proposition has been removed from the paper in the published paper, available in the author's webpage.