Extra symmetry of transfer operators for the dilute $A_3$ model

For the dilute $A_3$ model, the parameters are $L=3$, $r=8/5$, and $r^*=3/5$. Let $T_n(u)$ be the transfer operators acting on the BRST cohomology groups $H^0(C_{l,k})$ for $k=1,2,3$ and $l=1,2$.

Extra symmetry conjecture. As operators on these BRST cohomology groups,

$$T_3(u)=\frac{T_2(u)}{[\\![2/10]\\!][\\![3/10]\\!]},$$ $$T_4(u)=\frac{-T_1(u)}{[\\![-9/10]\\!][\\![-4/10]\\!][\\![-3/10]\\!][\\! [3/10]\\!][\\![2/10]\\!][\\![8/10]\\!]},$$ $$T_5(u)=\frac{-\operatorname{id}}{[\\![-15/10]\\!][\\![-10/10]\\!][\\![-9/10]\\!][\\![-4/10]\\!][\\![-3/10]\\!][\\![2/10]\\!][\\![3/10]\\!][\\![8/10]\\!][\\![9/10]\\!][\\![14/10]\\!]}.$$

The paper expects these identities as an additional symmetry specific to the dilute $A_3$ model, but the supplied text gives no proof or resolution. Their status is therefore open.