Powers of matrices with all principal minors equal to 1

Consider a square matrix $A$ whose all principal minors are equal to $1$. Over a field, this property is inherited by any power of $A$, but this is not the case over an arbitrary commutative ring. We show that it is the case over any regular ring, and also over the ring $\mathbb{Z} / d$ for any integer $d$, and in some other settings (quotients of Pr\"ufer domains and principal quotients of normal domains). This generalizes Problem B5 of the 2021 Putnam contest. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A = \left(a_{i,j}\right)_{i,j\in\left[n\right]}$ is strongly $1$-principled if all its diagonal entries are $1$ and if all the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ vanish. We show that the latter products are always integral over the ideal generated by the principal minors of $A$ minus $1$.