Klurman–Pohoata conjecture on large additive or multiplicative Sidon subsets

Let $A\boldsymbol{\subset}\mathbb R$ be finite. Define $s_+(A)$ as the size of the largest additive Sidon subset of $A$, and $s_*(A)$ as the size of the largest multiplicative Sidon subset of $A$. A set is additive Sidon if the only solutions of $a+b=c+d$ with $a,b,c,d$ in the set are the trivial ones, and multiplicative Sidon if the analogous condition holds for $ab=cd$.

Klurman–Pohoata conjecture. There exists a constant $c>0$ such that, for every finite $A\subset\mathbb R$,

$$\max\{s_+(A),s_*(A)\}\gg |A|^{1/2+c}.$$

The conjecture asks whether every finite real set contains a substantially larger-than-square-root additive or multiplicative Sidon subset. The source attributes it to Klurman and Pohoata and also records their stronger conjecture with exponent $1-\epsilon$ for every $\epsilon>0$; no resolution is supplied here.