The summability converse for spectral triples on the Sierpiński gasket

Let $SG$ denote the Sierpiński gasket and let $(A,H,D)$ be a spectral triple for $C(SG)$ inducing the geodesic distance on the gasket. Suppose that it is summable for every $p>\log 3/\log 2$, and that $(f(m,k))\in\Pi_{HSG}\mathbb{Z}$ represents its K-homology class. Summability converse. Then

$$\limsup_{m\to\infty}\left(\sum_{k=1}^{3^{m-1}}|f(m,k)|\right)^{1/m}\leq 3.$$

This would provide a converse to the preceding characterization of summability for the spectral triples $ST(f)$, extending the known implication from bounded sequences to a necessary growth condition for arbitrary spectral triples inducing the geodesic metric. The source does not state whether the converse is resolved.