Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I_n$ be the $n\times n$ identity matrix, and let $A_n$ be the $n\times n$ diagonal matrix having coefficient $1,q,...,q$ on the diagonal.

Note that $I_n$ and $A_n$ are always congruent over the $p$-adic integers for all $p$ not dividing $q$. Further assume that $I_n$ and $A_n$ are congruent over $\mathbf{Q}_p$ for $p$ dividing $q$ (which is the case when $q=10,12$ or $15$). Does it follow that $I_n$ is congruent to $A_n$ over $\mathbf{Z}[1/q]$ ?

$\textbf{Motivation}$: A friend explained the paper "On the non-existence of certain finite projective planes", and a naïve hope is to push a bit further their strategy by answering the above question in the negative. This has for sure already been tried, and I would like to know some references about that.

$\textbf{Edit}$:In view of the question What's in the genus of the cubic lattice?, the following question related to the above discussion might be easier to answer: the two quadratic forms over $\mathbf{Z}$ given by $\sum \limits_{i=1}^{111} x_i^2$ and $x_1^2+ \sum \limits_{i=2}^{111} 10x_i^2$ are in the same genus. But are they actually isomorphic over $\mathbf{Z}$? Can I hope for an answer from Magma, or would it be overwhelmed by the number of variables ?