QUESTION. Are my following conjectures true? How to prove them?

Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that $$\left(\frac ap\right)=\left(\frac bp\right)=\left(\frac cp\right)=1\ \mbox{and}\ a^2+b^2=c^2,$$ where $(-)$ denotes the Legendre symbol.

Conjecture 2. For each prime $p>50$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that $$\left(\frac ap\right)=\left(\frac bp\right)=\left(\frac cp\right)=-1\ \mbox{and}\ a^2+b^2=c^2.$$

Conjecture 3. For each prime $p>32$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that $$\left(\frac ap\right)=\left(\frac bp\right)=1,\ \left(\frac cp\right)=-1\ \mbox{and}\ a^2+b^2=c^2.$$

Conjecture 4. For each prime $p>72$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that $$\left(\frac ap\right)=\left(\frac bp\right)=-1,\ \left(\frac cp\right)=1\ \mbox{and}\ a^2+b^2=c^2.$$

Remark. I formulated these conjectures in 2015, see http://oeis.org/A260911 . Perhaps, it is practical to prove them.