Conjecture on Legendre-function GKN conditions and bases for left-definite domains

Let $P_{j_1},P_{j_2},\dots,P_{j_n}$ be any set of $n$ distinct Legendre polynomials, with $n_1$ odd indices and $n_2$ even indices such that $n_1+n_2=n$. Let $Q_{k_1},Q_{k_2},\dots,Q_{k_n}$ be $n$ distinct Legendre functions of the second kind, with $n_2$ odd indices and $n_1$ even indices.

Legendre-function conjecture. These Legendre polynomials can be used as GKN conditions to define the $n/2$ left-definite domain, and together the $2n$ functions constitute a basis of the space $\mathcal{D}_+^n\dotplus\mathcal{D}_-^n$.

This conjecture concerns the additional index conditions needed for the matrix governing the GKN conditions to have full rank. The preceding proposition establishes the necessary equality between the numbers of even and odd indices, but the source indicates that sufficiency and the general rank question remain unresolved.