Quadratic characters with non-negative partial sums

Are there infinitely many quadratic characters (for instance, the Legendre symbol mod p) for which the partial sums are always non-negative?   Although only 0% of characters can have this property, numerical work (most recently by Kalmynin) suggests that such characters are nevertheless plentiful.  For instance, computations show that there are many more examples than may be expected by modeling such sums by a simple random walk.   I will discuss joint work with Angelo and Xu which obtains new upper bounds for the number of such characters, by studying a related problem on maxima of L-values (closely connected to the Fyodorov-Hiary-Keating conjectures).   I will give a heuristic explanation for why such characters are plentiful, and which suggests that our upper bounds may not be too far from the truth.