Conjecture on global analytic lifts of weakly analytic maps

Let $Y$ be a $k$-affinoid space, let $X$ be a basic tube, and let $f:X\to Y$ be a weakly analytic map, meaning locally a map of the form $\pi_{K/k}\circ F\circ\sigma_{K/k}$ for a complete extension $K/k$ and a $K$-analytic map $F$. Here $X_K$ and $Y_K$ are the base changes to $K$, and $\sigma_{K/k}$ and $\pi_{K/k}$ are the associated maps between the base-changed and original analytic spaces. Global lifting conjecture for weakly analytic maps. There exist a complete extension $K/k$ and an analytic map $F:X_K\to Y_K$ such that

$$f=\pi_{K/k}\circ F\circ\sigma_{K/k}.$$

Weakly analytic maps are characterized locally as pointwise limits of analytic maps and share several analytic properties, but the source leaves this global lifting assertion open even on basic tubes.