Infinite log-convexity conjecture for $(1,1)$-Hoggatt sums

For a fixed integer $m\geq 2$, let $H_n^{[m]}(1,1)$ denote the $(1,1)$-Hoggatt sums. The $(1,1)$-Hoggatt-sum infinite log-convexity conjecture. For every fixed $m\geq 2$, the sequence

$$\{H_n^{[m]}(1,1)\}_{n\geq 1}$$

is infinitely log-convex. This generalizes the Baxter-number case, since the Baxter numbers are the specialization $H_n^{[3]}(1,1)$; the source presents it as a proposed generalization, and no proof is given.