Hardness conjecture for immanants of large-depth Young diagrams

Let $\lambda(n)$ be any family of Young diagrams, where the depth of $\lambda(n)$ is the number of rows in the diagram. Assume that, for some constant $\delta>0$, the depth is $n-n^\delta$. Immanant hardness conjecture. Then $\mathrm{Imm}_{\lambda(n)}$ is $\#\mathrm{P}$-hard. Equivalently, this concerns any family with a polynomial number of boxes to the right of the first column. The conjecture extends the paper's established hardness results for immanants associated with Young diagrams of bounded width, and its status is open.