The embedded Nash problem of birational models of rational triple singularities

We consider the question whether one can construct an embedded resolution of singularities of a singular variety $X\subset {\mathbf{\text{A}}}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular locus of $X\mathrm{.}$ We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.