I recently watched a video showing that it is possible to have the same polyomino fold into two distinct cuboids. Below is an example of such a cuboid (the purple bit is the net) of area $22$, which folds into a $1$x$1$x$5$ cuboid or a $1$x$2$x$3$ cuboid:
This example also happens to have the minimal possible surface area (though it is not the only minimal example).
The video continues that there exist nets that can be folded into three distinct cuboids. However, the smallest known examples either has area $532$, or fold along lines not parallel to the axes. The latter example, though elegant, and area $30$, does bend (pun intended) the rules.
I would like to know whether there is a smaller example (than $532$), which does fold along the grid lines you'd intuitively expect for a net.
A nice starting point into this problem is to consider the different possible surface areas that three different cuboids could simultaneously have. It turns out that the smallest possible surface area for three integer cuboids is $46$, the surface area of a $1$x$1$x$11$ cuboid, a $1$x$2$x$7$ cuboid, or a $1$x$3$x$5$ cuboid.
My question is therefore: Does there exist (and show examples if there is) a polyomino net of surface are $46$ which can be folded into those three cuboids?