Let $K$ be a complete field with respect to a discrete valuation $v$ and let $O_K$ be its valuation ring, $m$ its maximal ideal. Suppose $K$ has characteristic $0$ and that $O_K/m$ is of characteristic $p$.
Let $A$ be an $O_K$ algebra and suppose that $A$ is a complete regular local ring of dimension $d+1$. In general, I know that there exists a complicated result (Cohen Structure Theorem, unequal characteristic case) that asserts that $A$ is more or less a homomorphic image of a ring of power series in $d$ variables over a suitable ring of Witt vectors.
My question is: assume that $A\otimes_{O_K} K = K[[X_1, \ldots, X_d]]$ and that the square of the maximal ideal $m_A$ of $A$ does not contain the uniformizer $\pi$ of $O_K$ (in Cohen's language: A is unramified). Can we deduce in this case that $A=O_K[[X_1, \ldots, X_d]]$ (possibly without Cohen's theorem)?