The $a\equiv3\pmod 4$ Stirling valuation conjecture

Let $f(b,c)=\sigma_2(b)-\sigma_2(c)+\nu_2\left(\binom{2c-b-1}{c}\right)$, where $c3_2$ is the binary digit sum and $bd_2$ is the $2$-adic valuation. Let $a,b,c,h$ be positive integers satisfying $c>b\ge1$ and $3\le a\le2^{h-1}$ with $a\equiv3\pmod 4$. The $a\equiv3\pmod 4$ conjecture. One has

$$\nu_2\left(S\left(c2^h+2,b2^h+a\right)\right)=\sigma_2(a)+f(b,c)+\nu_2\left(\binom{a+1}{2}\right)-1.$$

The claim is motivated by numerical experimentation and extends the theorem's lower bound in the residue class $a\equiv3\pmod4$; its resolution is not given in the supplied text.