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• Given 4, 2 and 5 output 3, 2 cap the highest (cth), ko
• Given 2, 4 and 5 output 2, 3 cth, ko
• Given 3, 3 and 5 output 3, 2 cap any of the highest (cath)

• Given 4, 2 and 5 output 3, 2 cap the highest (cth), ko
• Given 2, 4 and 5 output 2, 3 cth, ko
• Given 3, 3 and 5 output 3, 2 or 2, 3 cap any of the highest (cath)

• Given 5, 4, 2 and 10 output 4, 4, 2 cth, ko
• Given 2, 4, 5 and 10 output 2, 4, 4 cth, ko
• Given 4, 4, 2 and 10 output 4, 4, 2 ncn, ko
• Given 4, 2, 4 and 10 output 4, 2, 4 ncn, ko
• Given 4, 4, 2 and 7 output 3, 2, 2 or 2, 3, 2 cth, cath, ko
• Given 4, 2, 4 and 7 output 3, 2, 2 or 2, 2, 3 cth, cath, ko

• Given 4, 4, 2 and 5 output 2, 2, 1 or any permutation (oap)
• Given 2, 4, 4 and 5 output 1, 2, 2 oap
• Given 4, 4, 2 and 4 output 1, 2, 1 oap
• Given 2, 4, 4 and 4 output 1, 1, 2 oap

• Given 5, 4, 2 and 10 output 4, 4, 2 cth, ko
• Given 2, 4, 5 and 10 output 2, 4, 4 cth, ko
• Given 4, 4, 2 and 7 output 3, 2, 2 or 2, 3, 2 cth, cath, ko
• Given 4, 2, 4 and 7 output 3, 2, 2 or 2, 2, 3 cth, cath, ko

• Given 4, 4, 2 and 5 output 2, 2, 1 or any permutation (oap)
• Given 2, 4, 4 and 5 output 1, 2, 2 oap
• Given 4, 4, 2 and 4 output 1, 2, 1 oap
• Given 2, 4, 4 and 4 output 1, 1, 2 oap
• Given 70, 80, 90 and 10 output 3, 3, 4 oap

• Given 4, 2 and 6 output 4, 2 no capping needed (ncn), keep order (ko)
• Given 2, 4 and 6 output 2, 4 ncn, ko
• Given 3, 3 and 6 output 3, 3 ncn

• Given 4, 4, 2 and 5 output 2, 2, 1 cth, ko
• Given 2, 4, 4 and 5 output 1, 2, 2 cth, ko
• Given 4, 4, 2 and 4 output 1, 2, 1 or 2, 1, 1 cth, cath, ko
• Given 2, 4, 4 and 4 output 1, 1, 2 or 1, 2, 1 cth, cath, ko

• Both input and output are all about positive integers (>0)
• The number of values in the input and output list are equal.
• The sum of the output values is exactly equal to m or less than m only if the sum of the input values was already lower.
• If the sum of the values in the input list is already lower then or equal to m, no capping is needed. (ncn)
• Cap the highest values first (cth)
• When multiple values are the highest value and equally high, it doesn't matter which you cap. (cath)
• The capped values in the output list have to be in the same order as their original values in the input list. (ko)
• If there's no solution output an error or a falsy value.

The shortest valid answer - measured in bytes - wins.

• Given 4, 2 and 6 output 4, 2 no capping needed (ncn), keep order (ko)
• Given 2, 4 and 6 output 2, 4 ncn, ko
• Given 3, 3 and 6 output 3, 3 ncn
• Given 3, 3 and 7 output 3, 3 ncn

• Given 4, 4, 2 and 5 output 2, 2, 1 or any permutation (oap)
• Given 2, 4, 4 and 5 output 1, 2, 2 oap
• Given 4, 4, 2 and 4 output 1, 2, 1 oap
• Given 2, 4, 4 and 4 output 1, 1, 2 oap

• Both input and output are all about positive integers (>0)
• The number of values in the input and output list are equal.
• The sum of the output values is exactly equal to m or less than m only if the sum of the input values was already lower.
• If the sum of the values in the input list is already lower then or equal to m, no capping is needed. (ncn)
• Cap the highest values first (cth)
• When multiple values are the highest value and equally high, it doesn't matter which you cap. (cath)
• The capped values in the output list have to be in the same order as their original values in the input list. (ko)
• When at some point (thinking iteratively) all values are equally high, it stops being important which you cap.
• If there's no solution output an error or a falsy value.

The shortest valid answer - measured in bytes - wins.

Apart form the rules, I'm interested to see a program that keeps a value intact as long as possible. Consider the input values a plant each with the heights 10,2,10 and the maximum m=5, it would be a waste to cap the baby plant in the middle.