The binary-representation conjecture for fractional parts of binomial coefficients

Let $p\neq 2$ be a prime, let $m=2p$, and let $m_lm_{l-1}\cdots m_1m_0$ be the standard base-$2$ representation of $m$. Let $n\geq 1$. Write $n_1$ for the leading digit in the base-$p$ representation of $n$, let $|m|_0$ denote the number of zeros in the standard binary representation of $m$, and let $\operatorname{mod}(a,b)$ be the unique integer $x$ with $x\equiv a\pmod b$ and $0\leq x<b$. Binary-representation conjecture. The quantity

$$\frac{2}{n}\binom{n}{2p}-\frac{(p+1)(n_1-1)}{p}\delta_{p\mid n}$$

is odd if and only if

$$n\equiv 2p+\sum_{i=0}^l\left\lfloor\frac{j}{2^{|\operatorname{mod}(2p,2^{i+1})|_0-\delta_{i=1}}}\right\rfloor m_i2^i\pmod{2^{\lfloor\log_2(2p)\rfloor+1}}$$

for some $0\leq j\leq 2^{|2p|_0}-1$. This criterion is intended to determine the fractional part of $\frac{1}{n}\binom{n}{2p}$ from the binary representation of $p$; its resolution is not given here.