The 2-adic shift conjecture for involution numbers

Let $t_{4k+1}^e$ denote the even involution count associated with size $4k+1$, let $\operatorname{ord}_2$ be the 2-adic valuation, and let $\chi_o(k)$ be the indicator of odd integers. A 2-adic integer is an element of $\mathbb{Z}_2$; write

$$\rho=\sum_{i\geq 0}\rho_i2^i,$$

where $0\leq\rho_i\leq1$.

2-adic shift conjecture. There is a 2-adic integer $\rho$ satisfying

$$\operatorname{ord}_2(t_{4k+1}^e)=k+\chi_o(k)\bigl(\operatorname{ord}_2(k+\rho)+1\bigr).$$

This conjecture gives the missing 2-adic valuation formula for the even involution counts $t_{4k+1}^e$; the paper presents it as an extrapolation from Maple experiments, and no proof or resolution is supplied.