Rippon's iterated exponential: are all Taylor coefficients of $\varphi_t^{n}(-1)$ bounded by $1$ in modulus? (Problem 7.54)

Let $\varphi_t(z)=e^{tz}-1$ and define the iterates $\varphi_t^{1}=\varphi_t$ and $\varphi_t^{n+1}=\varphi_t\circ\varphi_t^{n}$ for $n\ge1$. Each $\varphi_t^{n}(-1)$ is a formal power series in $t$; for example $\varphi_t^{1}(-1)=e^{-t}-1=-t+\tfrac{t^2}{2!}-\cdots$ and $\varphi_t^{2}(-1)=e^{t(e^{-t}-1)}-1=-t^2+\cdots$, and in general the substitution identity $\varphi_t^{n+1}(-1)=e^{t\cdot\varphi_t^{n}(-1)}-1$ gives an explicit rational recurrence for the coefficients. Question (Rippon): are the coefficients of these power series uniformly bounded by $1$ in modulus? Precisely, is it true that $\bigl|[t^k]\,\varphi_t^{n}(-1)\bigr|\le1$ for every $n\ge1$ and every coefficient index $k\ge0$?

Acceptance. FULLY RESOLVES: EITHER a proof that $|[t^k]\varphi_t^{n}(-1)|\le1$ for all $n\ge1$ and $k\ge0$ (e.g. a uniform inductive bound derived from the recurrence), OR an explicit pair $(n,k)$ together with the exact rational or algebraic value of $[t^k]\varphi_t^{n}(-1)$ of modulus strictly greater than $1$ (a self-certifying finite counterexample). ADVANCES: a re-runnable exact-arithmetic certificate confirming $|[t^k]\varphi_t^{n}(-1)|\le1$ for all $n$ and all $k$ up to a stated large bound, or a rigorous proof for a restricted family (all $k$ at fixed $n$, or a specified range of indices). Floating-point-only checks do NOT qualify.