Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be continuous. Do there then exist sequences $\boldsymbol c,\boldsymbol t\in I^{\ \mathbb N_0}$, $\boldsymbol c(i)=c_i$ and $\boldsymbol t(i)=t_i$, with

(1) $\quad\mathbb R\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^kc_i=1 \quad$ and

(2) $\quad E\ \text{-}\ \lim_{\ k\to\infty\ }\big(k^{-1}\sum_{i=0}^{k-1}\gamma(k^{-1}i)\big) = E\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^k(c_i\gamma(t_i)) \quad$ ?

Either a (sketch of a) proof of the positive case or a counterexample is welcome. Countable or σ−convexity has also been considered in this question.