Base-3 digit conjectures for nilpotency indices at 11 and 7 modulo 3

Let $D=\eta(2z)^{12}$, let $k\geq1$ with $6\nmid k$, write $k=\sum a_i3^i$, and let $s\equiv k\pmod{54}$. For $t\geq1$, define

$$w_t=N_{11}(D^{2\cdot3^t+1},3)-1,\qquad z_t=N_7(D^{2\cdot3^t+1},3)-1.$$

Base-3 digit conjecture.

$$N_{11}(D^k,3)=N_{11}(D^s,3)+\sum_{i\geq3}W_iw_i,$$

where the $W_i$ are linear combinations of $a_i$, $w_i=2w_{i-1}$ for $i\geq3$, and $w_2=2$; also

$$N_7(D^k,3)=N_7(D^s,3)+\sum_{i\geq3}Z_iz_i,$$

where the $Z_i$ are linear combinations of $a_i$, $z_i=z_{i-1}+2z_{i-2}$ for $i\geq3$, and $z_1=1,z_2=2$. Finally,

$$N_j(D^k,3)=O(k^{0.631})\quad(j\in\{7,11\}).$$

The conjecture concerns exact nilpotency indices for the two Hecke operators and gives a sublinear growth bound; the source reports computational verification only in a later remark.