Reference to a conjecture on unit vectors in Euclidean space

That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps:

1) Consider all sums $\sum_j \varepsilon_i u_i$ with $\varepsilon_i=\pm 1$ and choose the longest one. Replacing some $u_j$ with $-u_j$ if necessary, we can assume WLOG that it is $y=\sum_i u_i$. Comparing $y$ with $y-2u_i$ (a single sign flip) we get $$ \|y\|^2\ge \|y-2u_i\|^2=\|y\|^2-4\langle y,u_i\rangle+4\|u_i\|^2 $$ whence $\langle y,u_i\rangle\ge 1$ for all $i$. (That part is the main step in the solution of the plank problem).

2) Now we have $\|y\|^2=\sum_i\langle y,u_i\rangle\ge n$, so for $x=\frac y{\|y\|}$, we get $$ \sum_i\langle x,u_i\rangle=\sqrt{\sum_i\langle y,u_i\rangle}\ge \sqrt n $$ The End :-)