Let $B$ be a unital C*-algebra and $J$ is an ideal in it. Prove that if $x_1, \dots, x_n$ are positive elements of $B/J$ whose sum is $1$ then we can find positive lifts $\tilde{x}_1, \dots, \tilde{x}_n$ whose sum is $1$.
$x_1, \dots, x_n$ can be lifted to positive elements $\tilde{x}_1, \dots, \tilde{x}_n$. Let $s=\tilde{x}_1+ \dots + \tilde{x}_n$. If we can show that $s$ is positive we shall be able to prove that $s^{-1/2}\tilde{x}_1 s^{-1/2}, \dots, s^{-1/2}\tilde{x}_n s^{-1/2}$ are also positive lifts and their sum is $1$. I could not show that $s$ is invertible.
You have that there exists $a\in J$ such that $$\tag1 \tilde x_1+\cdots+\tilde x_n+a=1. $$ Since all other elements are selfadjoint, it follows that so is $a$. Now write $a=a_+-a_-$, with $a_+,a_-\in J^+$. Then $$ \tilde x_1+\cdots+\tilde x_n+a_+=1+a_-. $$ Now $s=1+a_-$ is positive and invertible, and $s+J=1+J$, so $s^{-1/2}+J=1+J$. Thus $$s^{-1/2}(\tilde x_1 +\tfrac1n\,a_+)s^{-1/2},\ldots,s^{-1/2}(\tilde x_n+\tfrac1n\,a_+) s^{-1/2} $$ are the desired lifts.