Powers of 2 in Balanced Grid Colourings

Let $B(m,n)$ be the number of ways to colour a $2m\times 2n$ grid in black and white so that, in each row and each column, half of the cells are white and half are black. Bhattacharya conjectured that the exponent of $2$ in the prime factorisation of $B(m,n)$ equals $s_2(m)s_2(n)$, where $s_2(x)$ denotes the number of $1$s in the binary expansion of $x$. We confirm this conjecture in some infinite families of special cases; most significantly, when $m$ is of the form either $2^k$ or $2^k+1$ and $n$ is arbitrary. The proof when $m=2^k+1$ is substantially more difficult, and in connection with it we develop some general techniques for the analysis of inequalities between binary digit sums.