Quadratic-growth extension of the convex-operator Beurling theorem

Let $A\in\mathcal{B}(\mathcal{H})$ be an expansive, convex analytic operator whose moment sequence has linear growth, meaning that there exist constants $c_1,c_2$ such that

$$\|A^nx\|^2\leq(c_1n+c_2)\|x\|^2$$

for every $x\in\mathcal{H}$. The Beurling-type theorem is the conclusion established for such operators in the preceding lemma. Quadratic-growth conjecture. The above result continues to hold when the linear growth condition is relaxed to quadratic growth. This proposes extending the known convex-operator result to moment bounds of quadratic rather than linear order.