Males and females need to pair or there will be no new sparrows. However opposite variants also need to pair or for behaviour reasons there will be no successful young.
For instance, wM-wF spend so much energy fighting with each other they fail to raise young.
The y variants are less aggressive but better with young and are the most popular mate choices. The y-variants are also monogomous while the w-variants are not.
The aggressive w-variants grab all the y-variants as partners. The result is that y-y pairs are rarely (never?) seen, even though they could produce young.
The article notes that yellow throats have different food strategies from white throats. This will also tend to reinforce alternate gender pairing.
An MF pair is going to bring in less food if both are going after the same sources (they might start fighting!) while if each has a different strategy the young are less vulnerable to food shortages should one strategy or another not produce at any given time.
The four gender outcome seems to be in a sustainable equilibrium. If one is favoured, say wM-yF, then these sparrows would become like most other birds, with males having one colour pattern and females another.
Instead, behaviours have somehow distributed themselves across genders so a successful couple needs to match both if they are to raise the next generation of young.