Yes, your initial strategy would be rendered irrelevant since all that is saying is that you constrain

$\dfrac{w_{1}}{w_{1}+w_{2}}+\dfrac{w_{2}}{w_{1}+w_{2}}=0$

and so your solution is undefined if $w_1,w_2>0$. One way you could make your strategy useful under a sector-neutral constraint is to change it into an optimization that minimizes the differences between actual weights and unconstrained weights, subject to the above constraint. e.g. Find

$\underset{w_1',w_2'}{\min} \left(w_1'-w_1\right)+\left(w_2'-w_2\right)$

subject to

$\dfrac{w_{1}}{w_{1}+w_{2}}+\dfrac{w_{2}}{w_{1}+w_{2}}=0$

Yes, your initial strategy would be rendered irrelevant since all that is saying is that you constrain

$w_1 + w_2 = 0$

and so your solution is undefined if $w_1,w_2>0$. One way you could make your strategy useful under a sector-neutral constraint is to change it into an optimization that minimizes the differences between actual weights and unconstrained weights, subject to the above constraint. e.g. Find

$\underset{w_1',w_2'}{\min} \left(w_1'-w_1\right)+\left(w_2'-w_2\right)$

subject to

$w_1 + w_2 = 0$