The known results about the stable irrationality of very general smooth Fano complete intersections $X^n\subset\mathbb P^{n+c}$ of dimension $n\geq 3$ and fixed type $(d_1,\ldots, d_c)$ pose the natural problem of understanding under which conditions there (might) exist rational examples of that type/dimension, e.g. index one; index two; quartic fourfolds and fivefolds; etc. After introducing the problem and after briefly recalling the state of the art, we shall present some results and geometric motivations pointing towards the statement of a possible solution.