‘Liable’ to is used to denote negative predictions or undesirable outcomes; ‘likely’ is used for happy or positive forecasts. Since the whole sentence means to point to out that the program is cost effective, a positive feature, we must use ‘likely’ rather than ‘liable’. Remove A, B and D. Between C and E, C brings out the intended meaning neatly
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Which is the correct participle : proven vs proved

I think , no one has proved the fact for the students that they are less likely to need remedial programs later
So , passive form is not necessary
Students proved the point and hence a conclusion can be drawn. This is only expressed it we write the sentence like -

Since children who undergo preschool education have proved to be less likely to need remedial programs later, preschool education programs are demonstrably cost effective.
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