Let $S$ be the spectrum of a discrete valuation ring $R$, with fraction field $K$, algebraically closed residue field $k$, denote by $\varpi$ an uniformizer and by $o\in S$ the closed point. Let $\delta:X\rightarrow S$ be a projective morphism such that $X_K$ is a smooth $K$-variety. Let $x\in X_k$ be a closed point. Suppose that for every $j$ and $i$ mapping to $x$ and $o$ respectively there exists a morphism $Spec(R/\varpi^2)\rightarrow X$ commuting with the diagram below
$$\require{AMScd} \begin{CD} Spec(k) @>{j}>> X\\ @VVV @VVV \\ Spec(R/\varpi^2) @>{i}>> S \end{CD}$$
Does this imply $\delta$ is (formally) smooth at $x$? (see https://stacks.math.columbia.edu/tag/02GZ for the definition of formal smoothness)