Noncyclotomic growth conjecture for solutions of q-difference equations

Let $K$ be the coefficient field, let ${\mathcal P}_f$ be its finite places, and let ${\mathcal C}$ be the cyclotomic places. Suppose that

$$y=\sum_{n\geq 0}y_nx^n\in K[[x]]$$

is a solution of a $q$-difference equation with coefficients in $K$. Noncyclotomic growth conjecture. Then

$$\sigma_{{\mathcal P}_f\smallsetminus{\mathcal C}}(y)=\limsup_{n\to\infty}\frac{1}{n}\sum_{v\in{\mathcal P}_f\smallsetminus{\mathcal C}}\log^+\left(\sup_{s\leq n}|y_s|_v\right)<\infty.$$

The statement is presented as the expected analogue for $G_q$-functions of the preceding result for $G$-modules; the source supplies no proof or resolution.