Erdős problem 1026

Let $x_1,\ldots,x_n$ be a sequence of distinct real numbers. Determine $$\max\left(\sum x_{i_r}\right),$$ where the maximum is taken over all monotonic subsequences.

This is as Erdős posed the problem in [Er71], which is rather ambiguous. Discussion between several users in the comments section has led to the following precise possible question, as posed by van Doorn:

What is the largest constant $c$ such that, for all sequences of $n$ real numbers $x_1,\ldots,x_n$, $$\max\left(\sum x_{i_r}\right) > (c-o(1))\frac{1}{\sqrt{n}}\sum x_i$$ (where again the maximum is taken over all monotonic subsequences)?

Cambie makes the stronger conjecture that if $x_1,\ldots,x_{k^2}$ are distinct positive real numbers with $\sum x_i=1$ then there is always a monotonic subsequence with sum at least $1/k$.

This stronger conjecture appears to have been first proved by Tidor, Wang, and Yang [TWY16], and is also implicit in work of Wagner [Wa17]. A proof was given and formalised by Aristotle (see the comments), with an alternative proof provided by Chan. In particular, this shows that $c=1$.