Ideal Waring's conjecture

For each integer $k\ge 2$, let $g(k)$ be the least integer such that every positive integer is a sum of at most $g(k)$ $k$th powers, and define

$$I(k)=2^k+\left\lfloor(3/2)^k\right\rfloor-2.$$

Ideal Waring conjecture. For every $k\ge 2$,

$$g(k)=I(k).$$

This is the conjectural exact form of the lower bound arising from the standard obstruction near $3^k$. The general assertion remains open, although many individual values of $k$ are known.