The superexponential coefficient-growth conjecture for Kazhdan–Lusztig polynomials

For $w\in\mathrm{S}_{n}$, let $\mathrm{P}_{w}$ be the first-row Kazhdan–Lusztig polynomial, and let $\mathrm{coeff}_{n}$ denote the maximal coefficient among these polynomials; let $\mathrm{ev}_{n}$ denote the corresponding maximal value at $\mathtt{v}=1$. Superexponential coefficient-growth conjecture.

$$\mathrm{coeff}_{n}\in\Omega(\gamma^{n})\quad\text{for all }\gamma\in\mathbb{R}_{>1}.$$

Since $\mathrm{ev}_{n}\geq\mathrm{coeff}_{n}$, the same superexponential lower bound is asserted for $\mathrm{ev}_{n}$. The evidence is computational, and the conjecture concerns the asymptotic growth of the largest coefficients and evaluations.