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The answer to the main question is yes !
The (The question has been answered by the commenterscommentators).
As a consequence, it is not possible to push the Bruck-Ryser theorem by looking at congruences over rings of the form $\mathbf{Z}[1/q]$.
The question has been answered by the commenters.
The answer to the main question is yes ! (The question has been answered by the commentators).
