Higher-order Stirling-number monotonicity and log-concavity conjectures

Let $S(n,k)$ denote the Stirling numbers of the second kind, and let $\mathfrak{S}_{1}(n,k)$ be the quantity defined by the paper's equation defining $\mathfrak{S}_{1}(n,k)$. Recursively define

$$\mathfrak{S}_{\ell+1}(n,k)=\mathfrak{S}_{\ell}^2(n,k-1)-\mathfrak{S}_{\ell}(n,k-2)\mathfrak{S}_{\ell}(n,k)$$

and, for $n\geq k\geq \ell+2$, define

$$\mathcal{S}_{\ell}(n,k)=\frac{\mathfrak{S}_{\ell+1}(n,k)}{\mathfrak{S}_{\ell}(n,k)}.$$

Higher-order Stirling-number conjecture. The following claims are valid: (1) for fixed $ell\in\mathbb{N}$ and $n\geq \ell+3$, the finite sequence $\{\mathfrak{S}_{\ell}(n,k)\}_{\ell+1\leq k\leq n}$ is logarithmically concave in $k$; (2) for fixed $n\geq k\geq 3$, the finite sequence $\{\mathfrak{S}_{\ell}(n,k)\}_{1\leq \ell\leq k-1}$ is strictly increasing in $\ell$; (3) for fixed $\ell\in\mathbb{N}$ and $n\geq k\geq \ell+1$, the sequence $\{\mathfrak{S}_{\ell}(n+m,k+m)\}_{m\geq 0}$ is strictly increasing in $m$; (4) for fixed $\ell\in\mathbb{N}$ and $k\geq \ell+1$, the sequence $\{\mathfrak{S}_{\ell}(n,k)\}_{n\geq k}$ is strictly increasing in $n$; (5) for fixed $n\geq k\geq \ell+2$, the sequence $\{\mathcal{S}_{\ell}(n+m,k+m)\}_{m\geq 0}$ is strictly increasing in $m$; and (6) for fixed $k\geq \ell+2$, the sequence $\{\mathcal{S}_{\ell}(n,k)\}_{n\geq k}$ is strictly increasing in $n$.