Bergeron's conjectures for link symmetric functions

Let $u,v$ be binary words of compatible lengths, let $L_v(x;q,t)$ be the link symmetric function, and let $\widetilde{L}_v(x;q,t)=(1-q)^{n-|v|}L_v(x;q,t)$. Let $\widetilde{H}_\mu$ denote modified Macdonald polynomials, with $\nabla$ and $\nabla^{-1}$ the Macdonald operators; brackets denote the Lie bracket, and operators are applied to $1$ when no argument is specified. Bergeron's conjectures.

$$L_{v0}=L_{1v}+qL_{0v},$$ $$L_{0^n}=\sum_{v\in\{0,1\}^k}q^{n-|v|}L_{v0^{n-k}},$$ $$t\left(L_{u011v}-L_{u101v}\right)=L_{u101v}-L_{u110v},$$ $$\widetilde{L}_{0^a1^b0^c}=\nabla p_{1^c}\nabla^{-1}\widetilde{H}_{1^b}\nabla p_{1^a},$$ $$L_{1^a01^b}=\frac{t^a-1}{t^{a+b}-1}\left[\nabla p_1\nabla^{-1},\widetilde{H}_{1^{a+b}}\right]+\widetilde{H}_{1^{a+b}}p_1.$$

Bergeron also observed the additional positivity conjecture that $L_v(x;q,1+t)$ is $e$-positive. These formulas propose recursive, operator-theoretic, and positivity structures for the link symmetric functions.