Boros–Moll infinite q-log-convexity conjecture

Let $P_n(x)$ be the Boros–Moll polynomials, whose coefficients form a polynomial sequence in $n$. A polynomial sequence is infinitely $q$-log-convex if every iterate under the operator

$$\mathcal{L}\{f_i(q)\}_{i\geq 0}=\{f_{i-1}(q)f_{i+1}(q)-f_i(q)^2\}_{i\geq 1}$$

has coefficientwise nonnegative polynomials. Boros–Moll's conjecture. The Boros–Moll polynomials are infinitely $q$-log-convex. This extends known finite-order $q$-log-convexity results; the stronger infinite property is stated as an open conjecture.