Conjecture on large four-genera of connected sums of non-torsion solvable knots

Let $K$ be an $h$-solvable knot which is not torsion in the knot concordance group $\mathcal{C}$. For each positive integer $n$, let $\#^n K$ denote the connected \sum of $n$ copies of $K$.

Large four-genus conjecture. The collection $\{\#^n K\}$ consists of $h$-solvable knots and contains knots with arbitrarily large smooth four-genera $g_4$.

This conjecture strengthens the result established in the paper for $h=2$, where 2-solvable knots with arbitrarily large four-genera are constructed. It predicts the same phenomenon for every $h$-solvable knot that represents a non-torsion element of the knot concordance group.