Generalized Sidon-set existence conjecture for perfect powers

Let $k\geq 1$, $h\geq 2$, and $\varepsilon>0$. A $B_h[g]$ set $A\subseteq(\mathbb{Z}^{+})^k$ is a set of $k$-th powers with counting function $A(x)$. Generalized Sidon-set conjecture. For every $k\geq 1$, $h\geq 2$, and $\varepsilon>0$, there exists a $B_h[g]$ set $A\subseteq(\mathbb{Z}^{+})^k$ such that

$$A(x)\gg x^{\min\left\{\frac{1}{k},\frac{1}{h}\right\}-\varepsilon}.$$

The exponent is optimal up to the factor $x^{-\varepsilon}$ because every such set satisfies $A(x)\ll x^{\min\{1/k,1/h\}}$. The conjecture was proved for $k=1$ and all $h\geq2$, and the cited results also imply the case $k=h=2$; the general assertion remains unresolved in the source.