Canonical heights and preperiodic points for weighted homogeneous families of polynomials

A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{weighted homogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is weighted homogeneous. We prove a lower bound on the canonical height, of the form \[\hat{h}_{f_t}(z)\geq \epsilon \max\{h_{\mathsf{M}_d}(f_t), \log|\operatorname{Norm}\mathfrak{R}_{f_t}|\},\] for values $z\in K$ which are not preperiodic for $f_t$. Here $\epsilon$ depends only on the number of places at which $f_t$ has bad reduction. For suitably generic morphisms $\varphi:\mathbb{P}^1\to \mathbb{P}^1$, we also prove an absolute bound of this form for $t$ in the image of $\varphi$ over $K$ (assuming the $abc$ Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).