Connected-sum conjecture for powers of three in Khovanov torsion

Let $T(2,3)$ denote the trefoil, let $(\sigma_1\sigma_2\sigma_3)^4\sigma_1\sigma_2$ denote the braid whose closure is used in the connected sum, and let $\#_m$ denote the $m$-fold connected sum operation. Connected-sum conjecture for powers of three. For every $m\in\mathbb{Z}^{+}$, the Khovanov homology of

$$T(2,3)\ \#_{m}\ (\sigma_1\sigma_2\sigma_3)^4\sigma_1\sigma_2$$

contains the torsion subgroups $\mathbb{Z}_3,\mathbb{Z}_9,\ldots,\mathbb{Z}_{3^m}$.

The conjecture proposes an infinite family with torsion orders that are powers of $3$ and was verified in the source up to $m=4$; its general case remains open.