Linear rank growth for points on Legendre elliptic curves over solvable extensions

Let $K$ be a number field, and let $E$ be a Legendre elliptic curve over $K$ with parameter $\lambda\in K$. For each even integer $n\geq 4$, consider the $n$ points constructed above from the roots of $X^n-\lambda$. Linear rank-growth conjecture. There exist infinitely many even integers $n\geq 4$ such that the conditions of Theorem main-nf-case hold for all these $n$ points, and these points generate a subgroup whose rank grows linearly in $n$. The assertion is presented as a weaker possible statement because the author does not know whether the stated conditions hold for all but finitely many even integers $n\geq 4$; its resolution would give rank growth for the constructed points over the corresponding finite solvable extensions.