Kollár–Villarino conjecture on finite Pythagoras numbers and even powers

Let $R$ be a commutative ring with identity. For a positive integer $m$, let $p_m(R)$ denote the least integer $k$ such that every element of $R$ expressible as a sum of $m$th powers is a sum of at most $k$ such powers, if such a bound exists; write $p_m(R)<\infty$ when this bound exists.

Kollár–Villarino conjecture. The following conditions are equivalent:

1. $p_2(R)<\infty$.
2. $p_{2n}(R)<\infty$ for some $n$.
3. $p_{2n}(R)<\infty$ for all $n$.

This conjecture concerns whether finiteness of the Pythagoras number is equivalent to finiteness of the numbers of summands required for every even power. The source attributes it to the third author and Vill; no resolution is given here.