Prefix sum recurrence conjecture for Legendre-symbol walks

Let $p$ be a prime and let $\chi$ be the Legendre symbol modulo $p$. For $a\in\mathbb{F}_p$ and $t\in\{0,1,\ldots,p-1\}$, define the prefix sum

$$\Psi(a,t)=\sum_{j=0}^{t}\chi(a+j).$$

Prefix sum recurrence conjecture. For an absolute constant $A>0$, independently of $a\in\mathbb{F}_p$,

$$\left|\{t:\Psi(a,t)=0\}\right|=O\left(\sqrt{p}(\log p)^A\right).$$

A stronger related question asks whether, for some absolute constant $A>0$,

$$\max_{h\in\mathbb{Z}}\left|\left\{0\leq t\leq p-1:\Psi(0,t)=h\right\}\right|=O\left(\sqrt{p}(\log p)^A\right).$$

These estimates would control how often the Legendre-symbol prefix-sum walk attains a given value, and hence the frequency of intersection numbers in the associated projective-plane construction. The stated status of these bounds is unknown.