I believe it would be the middle one, children's
The correct form of the sentence would be "[They] will work with the students on the program."
The opposite of genuine is: SPECIOUS
According to Google this are the definitions for the following words.
<u><em>Genuine:</em></u><em> </em>honest, authentic
<em>Solicitude:</em> caring for something or someone
<u><em>Specious:</em></u><em> </em>misleading in appearance, fake
<em>Abrogated: </em>evading, getting away with something
<em>Cursory: </em>hasty, not detailed
<em>Superlative: </em>a high comparison
<em>Vitiate: </em>to spoil the quality of something
<em>Bugbear:</em> scary being like a hobgoblin
<em>Propensity:</em> tendency towards something
<em>Insipid: </em>lacking flavor
Therefore, the opposite of <em>honest and authentic </em>would be fake or<em> misleading in appearance </em>
