Flawless 1-factorization conjecture for even orders

For an even integer $n$, let $f(n)$ be the maximum integer $k$ such that some $k$-regular graph $G$ on $n$ vertices admits a flawless 1-factorization, meaning a perfect 1-factorization in which every Hamiltonian cycle of $G$ is the union of two 1-factors. Flawless 1-factorization conjecture.

$$f(n)=3\quad\text{for even }n\geq 4.$$

The paper states that $f(n)\geq 3$ for every even $n\geq4$ and conjectures that this lower bound is always sharp. The supplied text gives no further resolution status.