Hypothetical Non-Conservative Static Electric Field

Suppose an electric field as depicted in the picture
$$E \propto y$$ $$E = ky$$

Now when we take a charge $+q$ along the path $ABCDA$, Then :
$$W_{AB} = qky_2L$$ $$W_{BC} = 0 \text{(Since, displacement is perpendicular to the field)}$$ $$W_{CD} = -qky_1L$$ $$W_{DA} = 0 \text{(Since, displacement is perpendicular to the field)}$$

Total work done :
$$W = W_{AB} + W_{BC} + W_{CD} + W_{DA}$$ $$W = qky_2L + 0 - qky_1L + 0$$ $$W = qkL (y_2 -y_1)$$

Now if $y_2 \neq y_1$, then $W \neq 0$ Which implies that such particular sort of electric field would act non-conservatively.

Since it can be easily asserted that such a field is impossible as it does not follow that static electric field must be conservative, I have done the derivative for a estabilised field of line charge here.