Graphing conjecture for low-degree homotopy groups of long links

Let $n$, $p$, and $m$ be integers with $m\geq 2$ and $1\leq p\leq n-3$. Let $\mathcal{L}_{m\cdot p}^{n}$ denote the space of long links with $m$ components of dimension $p$ in $\mathbb{R}^{n}$, let $\mathcal{K}_{p}^{n}$ denote the corresponding space of long knots, and let $b_{21}$ and $b_{31}$ be the indicated braid classes. For each pair or triple of components, use graphing and joining as in the statement, and let $S^{n-p-1}$ be the $(n-p-1)$-sphere. Graphing conjecture. A minimal generating set for

$$\pi_{2n-3p-3}\mathcal{L}_{m\cdot p}^{n}$$

is given by the $m$ inclusions of a generator of $\pi_{2n-3p-3}\mathcal{K}_{p}^{n}$; the result of graphing and then joining two components of $[b_{21},b_{31}]$ for every pair of components $(i,j)$ with $1\leq i<j\leq m$; the image under graphing of a minimal generating set of $\pi_{2n-2p-3}S^{n-p-1}$ for every pair $(i,j)$ with $1\leq i<j\leq m$; and, if $m\geq 3$, the result of graphing $[b_{21},b_{31}]$ for every triple $(i,j,k)$ with $1\leq i<j<k\leq m$. This conjecture proposes an explicit description of generators in terms of knot inclusions, graphing, joining, and sphere homotopy classes; the torsion subgroup is the remaining issue in determining these groups.