Infinite logconcavity conjecture for the sequence $\{d_{\ell}(m)\}$

Let $\{d_{\ell}(m)\}$ be the sequence described above, and define the operator

$$\mathfrak{L}\left(\{x_k\}\right)=\left\{x_k^2-x_{k-1}x_{k+1}\right\}.$$

A sequence is infinitely logconcave if $\mathfrak{L}^j(\{x_k\})$ is nonnegative for every $j\in\mathbb{N}$. Infinite logconcavity conjecture. The sequence $\{d_{\ell}(m)\}$ is infinitely logconcave.

Logconcavity of this sequence is known, including through ratio-monotonicity and the minimum conjecture, but infinite logconcavity is the stronger proposed property and is not resolved here.