Holes in tileable double polynomino

This question was communicated to me by Evgeniy Romanov.

Consider a connected polyomino $ P $ which can be completely tiled in two different ways: with disjointed $ 2 times $ 2 square tetraminoes, and with disjoint S-shaped tetraminoes (which we allow to reflect and to make an arbitrary rotation). Is it true that $ P $ can not be simply connected, that is to say $ P $ must contain a "hole" in it?

The smallest non-trivial example of $ P $ is a corner of four $ 2 times $ 2 squares:

..AA.
BAAA.
BB.CC
.DDCC
.DD ..

We can see that $ P $ can be tiled by S-forms as follows:

..AA.
BAAC.
BB.CC
.BDDC
.DD ..

A brute force shows that there is no counterexample corresponding to a $ 9 times $ 9 square.