Non-symmetric coefficient conjecture for order-dividing bijections from dihedral to cyclic groups

Let $D_{2n}=\langle s,r\mid s^2=r^n=1,\ srs=r^{-1}\rangle$ be a dihedral group and let $\mathbb{Z}_{2n}$ be the cyclic group of the same order. Suppose that

$$f(s^ar^b)=xa+yb$$

is an order-dividing bijective function, where $x,y\in\mathbb{Z}$. Non-symmetric coefficient conjecture. The function

$$f(s^ar^b)=ya+xb$$

is not an order-dividing bijective function.