Full-parameter properties conjecture for the discrete p-Hardy–Rellich–Birman weight

Let $\ell\in\mathbb{N}$ and $p>1$, and let $\tilde{\rho}^{(\ell,p)}$ be the weight defined in the source by the formulas for $\tilde{\rho}^{(\ell,p)}$ and $\tilde{\mathfrak{g}}^{(\ell,p)}$.

Full-parameter properties conjecture. The following assertions hold:

1. $\tilde{\rho}^{(\ell,p)}$ is an optimal discrete $p$-Hardy–Rellich–Birman weight.
2. $\tilde{\rho}^{(\ell,p)}$ improves pointwise upon the classical $p$-Birman weight.
3. For every $n\geq\ell$, $\tilde{\rho}_n^{(\ell,p)}$ has a power-series expansion in negative powers of $n$ whose coefficients are all non-negative.

These properties are known in the restricted cases $\ell=1$, $p>1$, or $\ell\in\mathbb{N}$, $p=2$. The conjecture extends them to all $\ell\in\mathbb{N}$ and $p>1$; the proof technique available in the restricted cases does not cover the full parameter range.