Given two naturals $s<t$. Is there always a square (or at least a bigger rectangle) that can be tiled with $s\times t$ rectangles in an irreducible way (i.e. any grid line splitting it cuts at least one of the $s\times t$ rectangles of the tiling)?

E.g. for $(s,t)=(1,2)$, it is well known that a $\underline{6\times 6}$ square has no irreducible tiling, but a $\underline{8\times 8}$ one does, and so do in fact all other rectangles with even area and sides bigger than $5\times 6$. I think I have seen a similar statement for $(s,t)=(2,3)$ somewhere, but can’t seem to find the article anymore. So:

What is known about the existence of such a rectangle for given $(s,t)$, and maybe even about lower/upper bounds for the sides of a minimal one?