Base-7 digit conjectures for nilpotency indices at 13 and 29 modulo 7

Let $k\geq1$ with $7\nmid k$, write $k=\sum a_i7^i$, and let $s$ satisfy $s\equiv k\pmod{98}$. For $t\geq1$, define

$$x_t=N_{13}(\Delta^{2\cdot7^t+1},7)-1,\qquad y_t=N_{29}(\Delta^{2\cdot7^t+1},7)-1.$$

Base-7 digit conjecture.

$$N_{13}(\Delta^k,7)=N_{13}(\Delta^s,7)+\sum_{i\geq2}X_ix_i,$$

and

$$N_{29}(\Delta^k,7)=N_{29}(\Delta^s,7)+\sum_{i\geq2}Y_iy_i,$$

where each $X_i$ and $Y_i$ is a linear combination of the base-$7$ digits $a_i$.

These conjectural formulas describe the nilpotency indices through the residue of $k$ modulo $98$ and its higher base-$7$ digits; the supplied text gives no resolution evidence.