This is a follow-up to my previous question *An explicit series representation for the analytic tetration with complex height*.

Recall the definition $(11)$ from there: $$t(z) = \sum_{n=0}^\infty \sum_{k=0}^n (-1)^{n-k} \, q^{\binom {n-k} 2} {z \brack n}_q {n \brack k}_q ({^k a}).\tag1$$ Let us introduce additional parameters $p,s\in\mathbb Z^+$ into this formula (changes are shown in red): $$t(\color{red}p,\color{red}s,z) = \sum_{n=0}^\infty \sum_{k=0}^n (-1)^{n-k} \, q^{\,\color{red}p \, \binom {n-k} 2} {(z-\color{red}s)/\color{red}p \brack n}_{q^{\color{red}p}} {n \brack k}_{q^{\color{red}p}} ({^{\color{red}p\,k+\color{red}s} a}).\tag2$$ Basically, it means that we build our function from a thinner and shifted sample of tetration values.

Conjecture 1.Values of integer parameters $p,s$ do not matter, i.e. $$\forall p,s\in\mathbb Z^+, \, t(p,s,z) = t(z).$$

Let us generalize the definition of $t(p,s,z)$ to real-valued parameters $p,s\in\mathbb R^+$ (note that we switch to continuous tetration $t(z)$ in the last factor): $$t_{\mathbb R}(p,s,z) = \sum_{n=0}^\infty \sum_{k=0}^n (-1)^{n-k} \, q^{\,p \, \binom {n-k} 2} {(z-s)/p \brack n}_{q^p} {n \brack k}_{q^p} \color{red}t(p\,k+s).\tag3$$

Conjecture 2.Values of real parameters $p,s$ do not matter, i.e. $$\forall p,s\in\mathbb R^+, \, t_{\mathbb R}(p,s,z) = t(z).$$

Can we prove these conjectures? If yes, it means that we do not need all values of the discrete tetration (represented by the factor $(^k a)$ in $(1)$) to build a formula for the analytic tetration, but a very thin (although still infinite) subset of them already contains enough information to exactly reconstruct its values at all points in between (including skipped integer points). Can we make a step further and find a way to construct an explicit series for the analytic tetration that relies only on a finite set of values of the discrete tetration?