Can multinomial theorem https://en.wikipedia.org/wiki/Multinomial_theorem be used in the conditional expectation in On the conditional expectation? I.e. can I expand:
$$ E\left(\left|\sum_{i=1}^n a_ir_i\right|^{2k} \big| \sum_{i=1}^nr_i=T\right)=\sum_{k_1+\ldots k_n=2k}\frac{(2k)!}{k_1!\ldots k_n!}a_1^{k_1} \ldots a_n^{k_n}E\left(r_1^{k_1}\ldots r_n^{k_n}\big | \sum_{i=1}^nr_i=T\right) $$
Even if the sum is negative, $$\left|\sum_{i=1}^n a_ir_i\right|^{2k} = \left(\left|\sum_{i=1}^n a_ir_i\right|^{2} \right)^k = \left(\left(\sum_{i=1}^n a_ir_i\right)^{2} \right)^k = \left(\sum_{i=1}^n a_ir_i\right)^{2k}.$$ Your equation follows by using the linearity of the conditional expectation.